JavaScript (ES7), 176 bytes

Expects a list of [numerator, denominator] pairs.

a=>eval("for(o=n=a.length,q=n**n;q--;)o[(p=a.map((_,j)=>a.filter((_,i)=>!(q/n**i%n^j))).filter(a=>a+a)).length]||p.some(b=>b.map(([p,q])=>[P=P*q+p*Q,Q*=q],P=0,Q=1)|P%Q)?o:o=p")

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Commented

This is a version without eval() for readability.

a => {                     // a[] = input list of fractions
  for(                     // main loop:
    o =                    //   o[] = output, initially not an array
    n = a.length,          //   n = length of a[]
    q = n ** n;            //   q = counter, initialized to n ** n
    q--;                   //   stop once q = 0 has been processed
  )                        //
  o[                       // test o[], using the length of p[]:
    ( p =                  //   p[] = partition
      a.map((_, j) =>      //   for each element at index j in a[]:
        a.filter((_, i) => //     for each element at index i in a[]:
          !(               //       keep this entry if
            q / n ** i % n //       floor(q / n ** i) mod n
            ^ j            //       is equal to j
          )                //
        )                  //     end of filter()
      ).filter(a => a + a) //   end of map(); remove empty slots
    ).length               //   get the final length of p[]
  ] ||                     // abort if p[] is longer than o[]
  p.some(b =>              // for each list of fractions b[] in p[]:
    b.map(([p, q]) =>      //   for each fraction [p, q] in b[]:
      [                    //     we add p/q to P/Q by:
        P = P * q + p * Q, //       updating the numerator P
        Q *= q             //       updating the denominator Q
      ],                   //
      P = 0, Q = 1         //     start with P/Q = 0/1
    )                      //   end of map()
    | P % Q                //   trigger some() if Q is not a divisor of P
  ) ? 0                    // end of some(); do nothing if truthy
    : o = p;               // otherwise, update o[] to p[]
                           // (implicit end of for)
  return o                 // return o[]
}                          //

Vyxal, 14 bytes

ṖvøṖÞf'Ṡ¨=⌊;ÞG

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Times out for anything more than 5 fractions if you're lucky. Don't even bother giving it the third and fourth test case.

Inputs as a list of fraction objects. Due to technical limitations, this is done with the a flag for convenience.

Explained

ṖvøṖÞf'Ṡ¨=⌊;ÞG # We'll call the length of the list N so we can see why this takes so long
Ṗ              # Permutations of the input - returns a list of N! items
 vøṖ           # To each of the N! permutations, get all possible partitions. Each permutation has 2^(N-1) partitions, so there are now N! * 2^(N-1) sublists but still N! items
    Þf         # Flatten the entire thing by one level. This iterates over each of those N! * 2^(N-1) sublists.
      '    ;   # From those partitions of permutations, keep only those where: (also iterating over each of those N! * 2^(N-1) items)
       Ṡ       #   Summating each partition
        ¨=⌊    #   is invariant under flooring
            ÞG # Get the longest remaining item.

Wolfram Language (Mathematica), 148 bytes

Last@Sort@(y=Flatten)[(a=#;(x=Select)[a~TakeList~#&/@x[Range@n~Tuples~#&~Array~(n=Length@a)~y~1,Tr@#==n&],Tr@Mod[Tr/@#,1]==0&])&/@Permutations@#,1]&

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05AB1E, 24 22 bytes

œ€.œ€`éʒε.EOT.òDïQ}P}θ

-2 bytes by taking the input as a list of strings "a/b" instead of list of pairs [a,b].

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Explanation:

œ            # Get all permutations of the (implicit) input-list
 €.œ         # Get all partitions of each permutation
    €`       # Flatten it one level down to a single list of partitions
      é      # Sort it by length (shortest to longest)
ʒ            # Filter this list of partitions by:
 ε           #  Map over each part of this partition:
  .E         #   Evaluate each inner string as Elixir to get the decimals
    O        #   Sum them together
     T.ò     #   Round it to 10 decimal values to fix floating point errors
        DïQ  #   Check that it's an integer:
        D    #    Duplicate
         ï   #    Cast the copy to an integer (floor for positive integers)
          Q  #    Check if the two are equal
 }P          #  After the map: check if all parts in this partition were truthy
}θ           # After the filter: pop and leave the last/longest valid partition
             # (which is output implicitly as result)

Python3, 227 bytes

def f(n):
 q,r=[(n,[[]])],[]
 for n,b in q:
  if[]==n:r+=[b[:-1]];continue
  q+=[(n[:i]+n[i+1:],b[:-1]+[U,[]]if(U:=b[-1]+[n[i]])and int(Z:=sum(a/b for a,b in U))==Z else b[:-1]+[U])for i in range(len(n))]
 return max(r,key=len)

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Charcoal, 92 bytes

≔Ｅ⪪Ｓ,⁺⟦ικ⟧Ｉ⪪ι/θ≔ΠＥθ§ι³ηＦθ⊞ι×÷η⊟ι⊟ιＷθ«≔ΦＥＸ²ＬθΦθ﹪÷κＸ²ν²∧κ¬﹪ΣＥκ§μ²ηι≔§ι⌕ＥιＬκ⌊ＥιＬκι≔⁻θιθ⟦⪫Ｅι§κ⁰,

Try it online! Link is to verbose version of code. Explanation:

≔Ｅ⪪Ｓ,⁺⟦ικ⟧Ｉ⪪ι/θ

Input the fractions and split them into numerator and denominator.

≔ΠＥθ§ι³η

Calculate a common denominator that will work for all of the fractions.

Ｆθ⊞ι×÷η⊟ι⊟ι

Calculate the equivalent numerator for each fraction.

Ｗθ«

Repeat until there are no more fractions to process.

≔ΦＥＸ²ＬθΦθ﹪÷κＸ²ν²∧κ¬﹪ΣＥκ§μ²ηι

Get all of the subsets of the fractions but filter out the empty set and any set whose sum of numerators isn't a multiple of the common denominator.

≔§ι⌕ＥιＬκ⌊ＥιＬκι

Find the shortest such subset.

≔⁻θιθ

Remove the shortest subset from the remaining fractions.

⟦⪫Ｅι§κ⁰,

Output the original fractions from the shortest subset.

Nekomata -t, 5 bytes

Oᵖ{ŜË

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Explanation:

Oᵖ{ŜË | -t: lasT result (longest, in this case)
------+----------------------------------------
O     | Partitions of input
 ᵖ{   | Such that:
   Ŝ  |  The sum of each sublist
    Ë |  Is an integer

Ë is actually the powOf2 function (returns \$2^n\$), but fails on non-integer inputs.