How can I replace a set of Nested For Loops with a Functional Code Alternative?

Update: A faster alternative:

foo = Boole @* LessThan[4] @* Length @* Union @* Transpose @* 
   Developer`ToPackedArray @* List;

c0 = Outer[foo, mt, mt, 1];

c0 == c

True

Original answer:

mt = Transpose[m];
c1 = Boole @ Outer[Length@Union@Transpose[{##}] < 4 &, mt, mt, 1];

c1 == c

True

Alternatively,

mt = Transpose[m];
c2 = ConstantArray[0, {32, 32}];
Do[c2[[i, j]] = c2[[j, i]] = Boole[Length@Union[Transpose[{mt[[i]], mt[[j]]}]] < 4], 
  {i, 1, Length @ columnlabels}, {j, 1, i}];

c2 == c

True

and

mt = Transpose[m];
c3 = SparseArray[{i_, j_} :> 
    Boole[Length@Union[Transpose[{mt[[ i]], mt[[j]]}]] < 4], 
  {1, 1} Length[columnlabels]]

Normal[c3] == c

True

and

mt = Transpose[m];
c4 = SymmetrizedArray[{i_, j_} :> 
   Boole[Length@Union@Transpose[{mt[[ i]], mt[[j]]}] < 4],
 {1, 1} Length[columnlabels], Symmetric[{1, 2}]]

Normal[c4] == c

 True

Using Table instead of For.

SeedRandom[99]
dat1 = Table[Table[RandomInteger[8], {6}], {6}];
dat2 = Table[Tally[dat1[[i]][[All]]], {i, 1, 6}];

wrapFn[x_List, yLimit_Integer] := 
 If[Length[x] < yLimit, 100, 0]

datOut = Table[wrapFn[dat2[[i]], 5], {i, 1, 6}]
(* Out: {100, 0, 100, 100, 100, 0} *)

Just if it helps the OP, with a pre-Fortran IV mind like myself? The O[?] is probably worse than the checked answer.

New Way:

SeedRandom[99]
dat3 = RandomInteger[8, {6, 6}]
dat4 = Map[Tally, dat3]
dat5Out = Map[If[Length[#] < 5, 100, 0] &, dat4]
(* Out: {100,0,100,100,100,0} *)

An improved version of kglr's answer, makes use of the fact that m only consists of 0 and 1:

m = RandomInteger[{0, 1}, {31, 2754}];

mt = Transpose[m];

func = Composition[Length, Union, Plus];

c2 = 1 - (Outer[func, mt, 2 mt, 1] - 4 // UnitStep); // AbsoluteTiming

(* {22.1601, Null} *)

kglr's solution takes about 53 seconds. Tested on v12.1, Wolfram Cloud.

Remark

My solution is slower in v9.0.1. (72 seconds v.s. 39 seconds. ) Not sure about the reason.

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Update

A solution with Compile (fastest one so far):

help = Compile[{{mat, _Integer, 2}}, 
   Table[If[4 > (lsti + 2 lstj // Union // Length), 1, 0], {lsti, mat}, {lstj, mat}](* , 
  CompilationTarget -> C *)]

test = help@mt; // AbsoluteTiming
(* {9.29816, Null} *)

If you have a C compiler installed, add the CompilationTarget -> C option and the code will be faster.

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P.S.

I didn't expect ContainsAll/SubsetQ is so slow.