spA = SparseArray[{{1, 1} -> 1, {2, 2} -> 2, {3, 3} -> 3, {1, 3} -> 4}]
    withOutDiagonalELements =
      DeleteCases[ArrayRules[spA], {a_, a_} -> _];
    spAwd=
      SparseArray@Append[withOutDiagonalELements, {_, _} -> 0.0];

spA = SparseArray[{{1, 1} -> 1, {2, 2} -> 2, {3, 3} -> 3, {1, 3} -> 4}];

zeros = 1 - IdentityMatrix[3, SparseArray];

spA*zeros // Grid

ReplacePart[spA, {i_, i_} :> 0]

Diagonal[ReplacePart[spA, {i_, i_} :> 0]] // Normal

ReplacePart[spA, {i_, i_} :> {a, b, c}[[i]]]