When you have two "contexts" (e.g. logic and presentation), build two separate families of functions. When doing computations with your structures, don't mess with their representations--let WL do the work. When you want to present your structures, define your own presentation functions. So, if you want a string, you need to determine rules for what a correctly formatted string representation of your structure is. From those rules, you can build the necessary functions. For example, if I just make guesses as to your presentation rules, I might come up with this:

TermString[term_Plus] := 
 StringRiffle[
  ToString@*InputForm /@ SortBy[List @@ term, FreeQ[x | y | z]], 
  " + "];
CoordinatesString[coords : {_, _, _}] := 
 StringRiffle[TermString /@ coords, {"{", ", ", "}"}]

Trying it out:

CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x}]
(* "{-y + 1/4, -z + 1/4, -x + 1/4}" *)

I haven't tested this much, and of course I don't know your exact presentation rules, so you'll probably need to adapt this.

I want to get the final result in string format.

I want to make it suitable for more variables, as shown below:

In[44]:= TermString[term_Plus] := 
 StringRiffle[
  ToString@*InputForm /@ SortBy[List @@ term, FreeQ[x | y | z|t|u|v]], 
  " + "];
CoordinatesString[coords : {_, _, _}] := 
 StringRiffle[TermString /@ coords, {"{", ", ", "}"}]

CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x}]
CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 2-t}]
CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2-t,2/3+u}]
CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2-t,2/3+u,4/5+v}]

Out[46]= "{-y + 1/4, -z + 1/4, -x + 1/4}"

During evaluation of In[46]:= StringRiffle::string: String expected at position 2 in StringRiffle[{-y + 1/4,-z + 1/4,-x + 1/4,-t + 2},{{,, ,, ,}}].

Out[47]= StringRiffle[{"-y + 1/4", "-z + 1/4", "-x + 1/4", 
  "-t + 2"}, {"{", ", ", ", ", "}"}]

Out[48]= CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 
  2/3 + u}]

Out[49]= CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 
  2/3 + u, 4/5 + v}]

It seems that the following method works well:

In[171]:= {{1/4 - y, 1/4 - z, 1/4 - x,1/2-t},{1/4 - y, 1/4 - z, 1/4 - x,1/2-t,3/4+u,5/6+v}}
(ToString[#[[2]],InputForm]<>"+"<>ToString[#[[1]],InputForm]&/@List@@@#)&/@%//StringRiffle[#,","]&//StringReplace[#,{"("->"",")"->"","},"->"\n","{"->"","}"->""}]&

Out[171]= {{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t}, {1/4 - y, 1/4 - z, 
  1/4 - x, 1/2 - t, 3/4 + u, 5/6 + v}}

Out[172]= "-y+1/4, -z+1/4, -x+1/4, -t+1/2
-y+1/4, -z+1/4, -x+1/4, -t+1/2, u+3/4, v+5/6"

However, when I further tried it on my actual data, I still didn't get the expected results:

gensFindGroup//ClearAll;
gensFindGroup[{m__?MatrixQ} | m__?MatrixQ]:=Block[{gens,dim,vec,vars,x,y,z,t,u,v,gensop},
  gens={m};
  dim=Dimensions[gens][[2]]-1;
  vars = {x, y, z, t, u, v};
  vec = Append[Take[vars, dim], 1];
  (*vec = {x, y, z, t, u, v}[[;; dim]]~Join~{1}*)

  (*Table[(i[[1;;3,1;;3]] . Transpose[{{x,y,z}}] + i[[1;;3,4]])//Transpose,{i,gens}]//ArrayFlatten[#,1]&//ToString[#,InputForm]&//StringReplace[#,{"("->"",")"->""}]&;*)
  (*Plus@@{ArrayFlatten[#[[1;;3,1;;3]] . Transpose[{{x,y,z}}],1],#[[1;;3,4]]}&/@gens// InputForm // ToString //StringReplace[#,{"("->"",")"->""}]&*)
gensop=Map[Dot[#, vec] &, gens[[All, 1 ;; dim]], {-2}];
(ToString[#[[2]],InputForm]<>"+"<>ToString[#[[1]],InputForm]&/@List@@@#)&/@gensop//StringRiffle[#,","]&//StringReplace[#,{"("->"",")"->"","},"->"\n","{"->"","}"->""}]&
]

In[318]:= "gensSGc2false:=[[[0,-1,0,1/4],[0,0,-1,1/4],[-1,0,0,1/4],[0,0,0,1]],[[-1,2,-1,-3/8],[-3/2,3/2,-1/2,-1/16],[-1/2,3/2,-3/2,-17/16],[0,0,0,1]]]"//StringReplace[#,{"["->"{","]"->"}"}]&//ToExpression
gensSGc2false//gensFindGroup

Out[319]= "-y+1/4, -z+1/4, -x+1/4
-x+-3/8, -3*x/2+-1/16, -1/2*x+-17/16"

The expected results should be as follows:

-y + 1/4, -z + 1/4, -x + 1/4 
-x + 2*y - z -3/8, - 3*x/2 + 3*y/2 - z/2 -1/16, - x/2 + 3*y/2 - 3*z/2 -17/16

Thank you very much, Eric,

As for your gensFindGroup, I have no idea where you're going with this. Please stop moving the goalposts.

Sorry, I digress too much. But the question discussed here is really a part of the question below.

Transform the affine matrix of space group elements to the desired generators required by Bilbao Crystallographic Server IDENTIFY GROUP , as shown below:

Based on your suggested approach, I present the following complete example:

In[228]:= TermString[term_Plus] := 
 StringRiffle[ToString@*InputForm /@ SortBy[List @@ term, NumericQ], 
  " + "];
CoordinatesString[coords : {__}] := 
 StringRiffle[TermString /@ coords, {"{", ", ", "}"}]

gens={{{0, -1, 0, 1/4}, {0, 0, -1, 1/4}, {-1, 0, 0, 1/4}, {0, 0, 0, 
   1}}, {{-1, 2, -1, -(3/8)}, {-(3/2), 3/2, -(1/2), -(1/16)}, {-(1/2),
    3/2, -(3/2), -(17/16)}, {0, 0, 0, 1}}};
dim=Dimensions[gens][[2]]-1;
vars = {x, y, z, t, u, v};
vec = Append[Take[vars, dim], 1];
Map[Dot[#, vec] &, gens[[All, 1 ;; dim]], {-2}];

CoordinatesString/@%;
StringReplace[#,{"("->"",")"->"","+ -"->"- "}]&/@%;
StringJoin[%]//StringReplace[#,{"}{"->"\n","{"->"","}"->""}]&

Out[237]= "-y + 1/4, -z + 1/4, -x + 1/4
-x + 2*y - z - 3/8, -3*x/2 + 3*y/2 - 1/2*z - 1/16, -1/2*x + 3*y/2 + \
-3*z/2 - 17/16"