Verbose, 1362 bytes

GET A ROMAN NUMERAL TYPED IN BY THE CURRENT PERSON USING THIS PROGRAM AND PUT IT ONTO THE TOP OF THE PROGRAM STACK
PUT THE NUMBER MMMM ONTO THE TOP OF THE PROGRAM STACK
MOVE THE FIRST ELEMENT OF THE PROGRAM STACK TO THE SECOND ELEMENT'S PLACE AND THE SECOND ELEMENT OF THE STACK TO THE FIRST ELEMENT'S PLACE
DIVIDE THE FIRST ELEMENT OF THE PROGRAM STACK BY THE SECOND ELEMENT OF THE PROGRAM STACK AND PUT THE RESULT ONTO THE TOP OF THE PROGRAM STACK
PUT THE NUMBER V ONTO THE TOP OF THE PROGRAM STACK
GET THE FIRST ELEMENT OF THE PROGRAM STACK AND THE SECOND ELEMENT OF THE PROGRAM STACK AND IF THE SECOND ELEMENT OF THE PROGRAM STACK IS NOT ZERO JUMP TO THE INSTRUCTION THAT IS THE CURRENT INSTRUCTION NUMBER AND THE FIRST ELEMENT ADDED TOGETHER'S RESULT
PUT THE NUMBER I ONTO THE TOP OF THE PROGRAM STACK
GET THE TOP ELEMENT OF THE STACK AND OUTPUT IT FOR THE CURRENT PERSON USING THIS PROGRAM TO SEE
PUT THE NUMBER III ONTO THE TOP OF THE PROGRAM STACK
GET THE FIRST ELEMENT OF THE PROGRAM STACK AND THE SECOND ELEMENT OF THE PROGRAM STACK AND IF THE SECOND ELEMENT OF THE PROGRAM STACK IS NOT ZERO JUMP TO THE INSTRUCTION THAT IS THE CURRENT INSTRUCTION NUMBER AND THE FIRST ELEMENT ADDED TOGETHER'S RESULT
PUT THE NUMBER NULLA ONTO THE TOP OF THE PROGRAM STACK
GET THE TOP ELEMENT OF THE STACK AND OUTPUT IT FOR THE CURRENT PERSON USING THIS PROGRAM TO SEE

Outputs I for valid roman numerals in the range I-MMMCMXCIX and NULLA (0) or informs user input is not a valid roman numeral otherwise.

C# (Visual C# Interactive Compiler), 79 109 bytes

This seems like a Regex challenge, I'm sure a shorter solution can be found...

s=>System.Text.RegularExpressions.Regex.IsMatch(s,"^M0,3(C[MD]|D?C0,3)(X[CL]|L?X0,3)(I[XV]|V?I0,3)$")

Try it online!

Wolfram Language (Mathematica), 35 bytes

Check[FromRomanNumeral@#<3999,1<0]&

Try it online!

5 bytes saved, thanks to @attinat

the limitation [1,3999] unfortunateley costs 7 bytes...

here is the code for any roman number

Wolfram Language (Mathematica), 28 bytes

Check[FromRomanNumeral@#,F]&

Try it online!

the above code works for any number, not just [1,3999]

CP-1610 assembly (Intellivision),  52 ... 48  47 DECLEs¹ = 59 bytes

Let's try this on a system that predates Perl by a good 7 years. :-)

Takes a pointer to a null-terminated string in R4. Sets the Zero flag if the input is a valid Roman numeral, or clears it otherwise.

 ROMW 10 ; use 10-bit ROM width
 ORG $4800 ; map this program at $4800

 ;; ------------------------------------------------------------- ;;
 ;; test code ;;
 ;; ------------------------------------------------------------- ;;
4800 EIS ; enable interrupts

4801 SDBD ; R5 = pointer into test case index
4802 MVII #ndx, R5
4805 MVII #$214, R3 ; R3 = backtab pointer
4807 MVII #11, R0 ; R0 = number of test cases

4809 loop SDBD ; R4 = pointer to next test case
480A MVI@ R5, R4
480B PSHR R0 ; save R0, R3, R5 onto the stack
480C PSHR R3
480D PSHR R5
480E CALL isRoman ; invoke our routine
4811 PULR R5 ; restore R5 and R3
4812 PULR R3

4813 MVII #$1A7, R0 ; use a white 'T' by default
4815 BEQ disp

4817 MVII #$137, R0 ; or a white 'F' is the Z flag was cleared

4819 disp MVO@ R0, R3 ; draw it
481A INCR R3 ; increment the backtab pointer

481B PULR R0 ; restore R0
481C DECR R0 ; and advance to the next test case, if any
481D BNEQ loop

481F DECR R7 ; loop forever

 ;; ------------------------------------------------------------- ;;
 ;; test cases ;;
 ;; ------------------------------------------------------------- ;;
4820 ndx BIDECLE test0, test1, test2, test3
4828 BIDECLE test4, test5, test6, test7, test8, test9, test10

 ; truthy
4836 test0 STRING "MCCXXXIV", 0
483F test1 STRING "CMLXXXVIII", 0
484A test2 STRING "DXIV", 0
484F test3 STRING "CI", 0

 ; falsy
4852 test4 STRING "MMIXVIII", 0
485B test5 STRING "IVX", 0
485F test6 STRING "IXV", 0
4863 test7 STRING "MMMM", 0
4868 test8 STRING "XXXVX", 0
486E test9 STRING "IVI", 0
4872 test10 STRING "VIV", 0

 ;; ------------------------------------------------------------- ;;
 ;; routine ;;
 ;; ------------------------------------------------------------- ;;
 isRoman PROC

4876 PSHR R5 ; push the return address

4877 MOVR R7, R2 ; R2 = dummy 1st suffix
4878 MOVR R2, R5 ; R5 = pointer into table
4879 ADDI #@tbl-$+1,R5

487B @loop MVI@ R5, R1 ; R1 = main digit (M, C, X, I)
487C MVI@ R5, R3 ; R3 = prefix or 2nd suffix (-, D, L, V)

487D MVI@ R4, R0 ; R0 = next digit

487E CMPR R0, R3 ; if this is the prefix ...
487F BNEQ @main

4881 COMR R2 ; ... disable the suffixes
4882 COMR R3 ; by setting them to invalid values
4883 MVI@ R4, R0 ; and read R0 again

4884 @main CMPR R0, R1 ; if R0 is not equal to the main digit,
4885 BNEQ @back ; assume that this part is over

4887 MVI@ R4, R0 ; R0 = next digit
4888 CMPR R0, R1 ; if this is a 2nd occurrence
4889 BNEQ @suffix ; of the main digit ...

488B CMP@ R4, R1 ; ... it may be followed by a 3rd occurrence
488C BNEQ @back

488E MOVR R2, R0 ; if so, force the test below to succeed

488F @suffix CMPR R0, R2 ; otherwise, it may be either the 1st suffix
4890 BEQ @next
4892 CMPR R0, R3 ; or the 2nd suffix (these tests always fail
4893 BEQ @next ; if the suffixes were disabled above)

4895 @back DECR R4 ; the last digit either belongs to the next
 ; iteration or is invalid

4896 @next MOVR R1, R2 ; use the current main digit
 ; as the next 1st suffix

4897 SUBI #'I', R1 ; was it the last iteration? ...
4899 BNEQ @loop

489B CMP@ R4, R1 ; ... yes: make sure that we've also reached
 ; the end of the input

489C PULR R7 ; return

489D @tbl DECLE 'M', '-' ; table format: main digit, 2nd suffix
489F DECLE 'C', 'D'
48A1 DECLE 'X', 'L'
48A3 DECLE 'I', 'V'

 ENDP

How?

The regular expression can be rewritten as 4 groups with the same structure, provided that # is any invalid character that is guaranteed not be present in the input string.

 +-------+---> main digit
 | |
(M[##]|#?M0,3)(C[MD]|D?C0,3)(X[CL]|L?X0,3)(I[XV]|V?I0,3)
 || |
 |+--+-----> prefix or second suffix
 |
 +---------> first suffix

The first suffix of the group $N$ is the main digit of the group $N-1$. Therefore, we can store the patterns with the pair $(textmain_digit, textsecond_suffix)$ alone.

Our routine attempts to parse the input string character by character according to these patterns and eventually checks whether the end of the string is reached.

Output

^{screenshot of jzIntv}

_{1. A CP-1610 opcode is encoded with a 10-bit value, known as a 'DECLE'. This routine is 47 DECLEs long, starting at $4876 and ending at $48A4 (included).}

Java 8, 70 bytes

s->s.matches("M0,3(C[MD]|D?C0,3)(X[CL]|L?X0,3)(I[XV]|V?I0,3)")

Port of @Innat3's C# answer, so make sure to upvote him!

Try it online.

Explanation:

s-> // Method with String parameter and boolean return-type
 s.matches("...") // Check if the string matches the regex fully
 // (which implicitly adds a leading "^" and trailing "$")

M0,3 // No, 1, 2, or 3 adjacent "M"
( | ) // Followed by either:
 C[MD] // A "C" with an "M" or "D" after it
 | // or:
 D? // An optional "D"
 C0,3 // Followed by no, 1, 2, or 3 adjacent "C"
( | ) // Followed by either:
 X[CL] // An "X" with a "C" or "L" after it
 | // or:
 L? // An optional "L"
 X0,3 // Followed by no, 1, 2, or 3 adjacent "X"
( | ) // Followed by either:
 I[XV] // An "I" with an "X" or "V" after it
 | // or:
 V? // An optional "V"
 I0,3 // Followed by no, 1, 2, or 3 adjacent "I"

R, 74 71 56 bytes

Thanks to @RobinRyder, @Giuseppe, & @MickyT for their suggestions how to use grep effectively with R's built in as.roman.

sub("^M(.+)","\1",scan(,""))%in%paste(as.roman(1:2999))

Try it online!

Wolfram Language (Mathematica), 32 bytes

RomanNumeral@Range@3999~Count~#&

Try it online!

Jelly,  48 47 46  44 bytes

-1 thanks to Nick Kennedy

5Żo7;“ÆæC‘ð“IVXLCDM”ṃ@3Ƥm2”MẋⱮ3¤ṭŻ€ṚŒpF€ḟ€0ċ

A monadic Link accepting a non-empty list of characters consisting only of IVXLCDM which yields either 1 (when it's a valid Roman numeral between $1$ and $3999$) or 0 (if not).

Try it online! Or see the test-suite.

How?

5Żo7;“ÆæC‘ð“IVXLCDM”ṃ@3Ƥm2”MẋⱮ3¤ṭŻ€ṚŒpF€ḟ€0ċ - Main Link: list of characters S

5Żo7;“ÆæC‘ - chain 1: f(S) -> X
5Ż - zero range of five = [0,1,2,3,4,5]
 o7 - OR seven [7,1,2,3,4,5]
 “ÆæC‘ - list of code-page indices [13,22,67]
 ; - concatenate [7,1,2,3,4,5,13,22,67]

 ð - start a new dyadic chain...

“IVXLCDM”ṃ@3Ƥm2”MẋⱮ3¤ṭŻ€ṚŒpF€ḟ€0ċ - chain 2: f(X,S) -> isValid
“IVXLCDM” - list of characters, IVXLCDM
 3Ƥ - for infixes of length three:
 - (i.e. IVX VXL XLC LCD CDM)
 ṃ@ - base decompression with swapped arguments
 - (i.e. use characters as base-3 digits of X's values)
 - (e.g. IVX -> VI I V IX II IV III VII VIII)
 m2 - modulo two slice (results for IVX XLC and CDM only)
 ¤ - nilad followed by link(s) as a nilad:
 ”M - character 'M'
 Ɱ3 - map across [1,2,3] with:
 ẋ - repeat -> M MM MMM
 ṭ - tack
 Ż€ - prepend a zero to each
 Ṛ - reverse
 - -- now we have the table: 
 - 0 M MM MMM
 - 0 DC C D CM CC CD CCC DCC DCCC
 - 0 LX X L XC XX XL XXX LXX LXXX
 - 0 VI I V IX II IV III VII VIII
 Œp - Cartesian product [[0,0,0,0],...,["M","CM",0,"IV"],...]
 F€ - flatten €ach [[0,0,0,0],...,['M','C','M',0,'I','V'],...]
 ḟ€0 - filter out the zeros from €ach ["",...,"MCMIV",...]
 ċ - count occurrences of S

Perl 5 (-p), 57 bytes

$_=/^M*(C[MD]|D?C*)(X[CL]|L?X*)(I[XV]|V?I*)$/&!/(.)13/

TIO

• uses almost the same regular expression except 0,3 quantifier was changed by *
  


• 
  &!/(.)13/ to ensure the same character can't occur 4 times in a row.


• can't be golfed with -/(.)13/ because would give-1 for IIIIVI for example

Python 2, 81 bytes

import re
re.compile('M,3(D?C,3|C[DM])(L?X,3|X[LC])(V?I,3|I[VX])$').match

Try it online!

Let's look at the last part of the regex, which matching the Roman numerals up to 9 (including the empty string)

V?I,3|I[VX]

This has two alternatives separated by |:

• 
  V?I,3: An optional V followed by up to 3 I's. This matches the empty string I,II,III, V, VI,VII,VIII.


• 
  I[VX]: An I followed by a V or X. This matches IV and IX.

The same things with X,L,C matching the tens, with C,D,M matches the hundreds, and finally ^M,3 allows up to 3 M's (thousands) at the start.

I tried generating the template for each trio of characters rather than writing it 3 times, but this was a lot longer.

Retina, 56 51 bytes

(.)13
0
^M*(C[MD]|D?C*)(X[CL]|L?X*)(I[XV]|V?I*)$

Port of @NahuelFouilleul's Perl 5 answer, so make sure to upvote him!

Try it online or verify all test cases.

Explanation:

(.)13 # If four adjacent characters can be found which are the same
0 # Replace it with a 0

^...$ # Then check if the string matches the following fully:
 M* # No or any amount of adjacent "M"
 ( | ) # Followed by either:
 C[MD] # A "C" with an "M" or "D" after it
 | # or:
 D? # An optional "D"
 C* # Followed by no or any amount of adjacent "C"
 ( | ) # Followed by either:
 X[CL] # An "X" with a "C" or "L" after it
 | # or:
 L? # An optional "L"
 X* # Followed by no or any amount of adjacent "X"
 ( | ) # Followed by either:
 I[XV] # An "I" with an "X" or "V" after it
 | # or:
 V? # An optional "V"
 I* # Followed by no or any amount of adjacent "I"

05AB1E, 61 9 8 bytes

ŽF¯L.XIå

Whopping $colorgreentextrm-52 bytes$ thanks to @Adnan, because apparently 05AB1E's Roman Number builtin wasn't documented, haha.. xD

Try it online or verify all test cases.

Explanation:

ŽF¯ # Push comressed integer 3999
 L # Create a list in the range [1,3999]
 .X # Convert each integer in this list to a roman number string
 Iå # Check if the input is in this list
 # (and output the result implicitly)

See this 05AB1E tip of mine (section How to compress large integers?) to understand why ŽF¯ is 3999.

Original 61 bytes answer:

•1∞Γ'иÛnuÞ₂…•Ž8вв€SÐ)v.•6#&‘нδ•u3ôNèyè}'M3L×)Rεõš}`3Fâ}€˜JIå

Try it online or verify all test cases.

Explanation:

•1∞Γ'иÛnuÞ₂…• '# Push compressed integer 397940501547566186191992778
 Ž8в # Push compressed integer 2112
 в # Convert the integer to Base-2112 as list:
 # [1,11,111,12,2,21,211,2111,10]
€S # Convert each number to a list of digits
 Ð # Triplicate this list
 ) # And wrap it into a list of lists (of lists)
 v # Loop `y` over each these three lists:
 .•6#&‘нδ• # Push compressed string "xivcxlmcd"
 u # Uppercased
 3ô # And split into parts of size 3: ["XIV","CXL","MCD"]
 Nè # Use the loop index to get the current part
 yè # And index the list of lists of digits into this string
 }'M '# After the loop: push "M"
 3L # Push list [1,2,3]
 × # Repeat the "M" that many times: ["M","MM","MMM"]
 ) # Wrap all lists on the stack into a list:
 # [[["I"],["I","I"],["I","I","I"],["I","V"],["V"],["V","I"],["V","I","I"],["V","I","I","I"],["I","X"]],[["X"],["X","X"],["X","X","X"],["X","L"],["L"],["L","X"],["L","X","X"],["L","X","X","X"],["X","C"]],[["C"],["C","C"],["C","C","C"],["C","D"],["D"],["D","C"],["D","C","C"],["D","C","C","C"],["C","M"]],["M","MM","MMM"]]
 R # Reverse this list
 εõš} # Prepend an empty string "" before each inner list
 ` # Push the four lists onto the stack
 3F # Loop 3 times:
 â # Take the cartesian product of the two top lists
 }€˜ # After the loop: flatten each inner list
 J # Join each inner list together to a single string
 Iå # And check if the input is in this list
 # (after which the result is output implicitly)

See this 05AB1E tip of mine (sections How to compress strings not part of the dictionary?, How to compress large integers?, and How to compress integer lists?) to understand why:

• 
  •1∞Γ'иÛnuÞ₂…• is 397940501547566186191992778
  


• 
  Ž8в is 2112
  


• 
  •1∞Γ'иÛnuÞ₂…•Ž8вв is [1,11,111,12,2,21,211,2111,10]
  


• 
  .•6#&‘нδ• is "xivcxlmcd"
  

perl -MRegexp::Common -pe, 34 bytes

$_=/^$REnumroman$/&!/(.)13/

The &!/(.)13/ part is necessary, because Regexp::Common allows four (but not five) of the same characters in a row. That way, it matches roman numbers used on clock faces, where IIII is often used for 4.

Python 3, 116 113 109 107 105 106 bytes

import re
lambda n:re.match(r'(M,3(C(M|CC?|D)?|DC,3))(X(C|XX?|L)?|(LX,3))?(I(X|II?|V)?|VI,3)?$',n)

Try it online!

-1 byte thanks to ShadowRanger

Ruby, (-n) 56 bytes

p~/^M,3(D?C,3|CM|CD)(L?X,3|XC|XL)(V?I,3|IV|IX)$/

Try it online!

Outputs 0 (truthy) or nil (falsy).