Manipulate a list: replace random position of a specific integer n times by 0

MapAt[0 &, #, RandomSample[Position[#, 1], UpTo@3]] & /@ d

{{0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}

Module[{s= SparseArray[RandomSample[Position[#, 1], UpTo@3] -> 0, Length@#, 1]}, s #] & /@ d

{{0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}

SubsetMap[{0, 0, 0} &, #, RandomSample[Position[#, 1], UpTo@3]] & /@ d

{{1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}

A function that takes a list and replaces a random 1 by a 0:

replaceone[L_] := ReplacePart[L, RandomChoice[Position[L, 1]][[1]] -> 0]

Apply it three times to all sublists of d:

Nest[replaceone, #, 3] & /@ d
(*    {{0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0},
       {0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1},
       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}    *)

Use ReplacePart:

d = {{1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0}, {0, 1, 1, 1, 0, 0, 1, 0, 1, 1,
    1}, {0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0}};

f = ReplacePart[#, RandomSample[Position[#, 1], 3] -> 0]&

Print[f /@ d]
(* {{1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}} *)

Try it online!

If you want it to work on lists that contain fewer than 3 1s, replacing them all with 0s, use UpTo[3] instead of 3.