What are my dimensions?

Jelly, 17 16 bytes

-1 byte thanks to Erik the outgolfer (make use of the quick, ¥)

SHð;_P
ṗ3Ç⁼¥Ðf²Ḣ

Brute force application of Heron's formula.

Try it online! (reaches the 60s time out for the 114 tests case. Takes 3m 30s locally - it does check 114³ = 1,481,544 triples)

How?

A true golf solution - given an area a it finds all tuples of three integers between 1 and a (even with repeated triangles and ones of no area), gets their area and filters for those with the desired area (it doesn't even stop as soon as one is found, it ploughs through them all and pops the first result afterwards). Yields 0 if none exists.

SHð;_P - Link 1, get the square of the area of a triangle: list of sides
S      - sum the sides (get the perimeter)
 H     - halve
  ð    - dyadic chain separation (call that p)
    _  - subtraction (vectorises) =    [p-side1,  p-side2,  p-side3]
   ;   - concatenate              = [p, p-side1,  p-side2,  p-side3]
     P - product                  =  p*(p-side1)*(p-side2)*(p-side3)
                                  = the square of Heron's formula = area squared

ṗ3Ç⁼¥Ðf²Ḣ - Main link: number a (area)
ṗ3        - third Cartesian power (all triples of [1,area] : [[1,1,1],[1,1,2],[1,2,1],[1,2,2],[2,1,1],[2,1,2],[2,2,1],[2,2,2], ... ,[a,a,a]]
       ²  - square a
     Ðf   - filter keep if:
    ¥     -   last two links as a dyad:
  Ç       -     call last link (1) as a monad f(list of sides)
   ⁼      -     left (that result) equals right (square of a)?
        Ḣ - head - get the first one (an empty list yields 0, perfect for the falsey case)

JavaScript (ES7), 109 102 100 98 bytes

Returns either an array of 3 integers or false. Like the Jelly answer, this is brute forcing Heron's formula.

A=>[...Array(A**3)].some((_,a)=>A*A/(r=[b=a/A%A|0,c=a/A/A|0,a%=A],p=a+b+c>>1)/(p-a)/(p-b)==p-c)&&r

Test cases

let f =

A=>[...Array(A**3)].some((_,a)=>A*A/(r=[b=a/A%A|0,c=a/A/A|0,a%=A],p=a+b+c>>1)/(p-a)/(p-b)==p-c)&&r

console.log(JSON.stringify(f(6)))
console.log(JSON.stringify(f(24)))
console.log(JSON.stringify(f(114)))
console.log(JSON.stringify(f(7)))

Recursive version, 83 bytes

Returns an array of 3 integers or throws a recursion error. Sadly, it only works for small inputs.

f=(A,n)=>A*A/(r=[a=n%A,b=n/A%A|0,c=n/A/A|0],p=a+b+c>>1)/(p-a)/(p-b)==p-c?r:f(A,-~n)

Demo

f=(A,n)=>A*A/(r=[a=n%A,b=n/A%A|0,c=n/A/A|0],p=a+b+c>>1)/(p-a)/(p-b)==p-c?r:f(A,-~n)

console.log(JSON.stringify(f(6)))
console.log(JSON.stringify(f(24)))

Haskell, 69 bytes

f a=take 1[t|t<-mapM(\_->[1..a])":-)",a*a==product[sum t/2-x|x<-0:t]]

Try it online!

Outputs a singleton of a list of three triangle sides like [[3.0,4.0,5.0]]. Impossible inputs give []. Technically only False is Falsey for Haskell, but because Haskell requires all possible outputs to be of the same type, it can't be used. If an error could be used as Falsey, [...]!!0 would save 3 bytes over take 1[..].

Tries all triples t of possible side lengths each ranging from 1 to the area a. Heron's formula is used to check if the area matches via (s-0)(s-x)(s-y)(s-z)==a*a where s=(x+y+z)/2 is sum t/2. The product (s-0)(s-x)(s-y)(s-z) is expressed as a product with elements taken from 0:t, i.e. the triple as well as 0.