Replace diagonal elements in sparse matrix

You can multiply a binary matrix by your Sparse Array, and that matrix may itself be a SparseArray.

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

zeros = 1 - IdentityMatrix[Dimensions @ spA, SparseArray];

spA*zeros // Grid

$\begin{array}{ccc} 0 & 0 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array}$

See also Band and DiagonalMatrix for other tools to construct the zeros array.

Just subtract the diagonal matrix:

SparseArray[spA - DiagonalMatrix[Diagonal[spA]]]

As Mr. Wizard pointed out, if spA is not a square matrix, one can use

k = 0;
SparseArray[spA - DiagonalMatrix[Diagonal[spA, k], k, Dimensions[spA]]]

Here, k stands for the number of the diagonal you what to delete. k = 0 stands for the main diagonal, k = 1 for the first diagonal above the main diagonal, k = -1 for the first diagonal below the main diagonal etc.

removeDiagonal = 
  # SparseArray[SparseArray`SparseArrayRemoveDiagonal[#]["NonzeroPositions"] -> 1, 
   Dimensions[#]] &;

removeDiagonal[spA]

SparseArray[<4>,{3,3}]

removeDiagonal[spA] // MatrixForm // TeXForm

$\left( \begin{array}{ccc} 0 & 0 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right)$

removeDiagonal[spA]["NonzeroPositions"]

{{1,3}}

Some timings:

f1 = removeDiagonal;
f2 = SparseArray[# - DiagonalMatrix[Diagonal[#, 0], 0,  
      Dimensions[#]]] &; (* Henrik Shumacher *)
f3 = With[{zeros = 1 - IdentityMatrix[Dimensions@#, SparseArray]}, #*
     zeros] &; (* Mr. Wizard *)
f4 = ReplacePart[#, {i_, i_} :> 0] &; (* tomd *)

m1 = SparseArray[Tuples[RandomSample[Range[101], 100], {2}] -> 1, {101, 101}];
m2 = SparseArray[Tuples[RandomSample[Range[100000], 100], {2}] -> 1, {100000, 100000}];

m = m1;
t11 = First[RepeatedTiming[r11 = f1 @ m ;]];
t21 = First[RepeatedTiming[r21 = f2 @ m;]];
t31 = First[RepeatedTiming[ r31 = f3 @ m;]];
t41 = First[RepeatedTiming[ r41 = f4 @ m;]];
r11 == r21 == r31 == r41

True

m = m2;
t12 = First[RepeatedTiming[r12 = f1 @ m ;]];
t22 = First[RepeatedTiming[r22 = f2 @ m;]];
t32 = First[RepeatedTiming[ r32 = f3 @ m;]];
t42 = "n/a"; (* b/c computation exceeded the limitation of free plan on  Wolfram Cloud *)
r12 == r22 == r32

True

{{"dimensions", "non-zero elements", "f1", "f2", "f3", "f4"},
 {{101, 101}, 100, t11, t21, t31, t41}, 
 {{100000, 100000}, 100, t12, t22, t32, t42}} // Grid[#, Dividers -> All] &  // TeXForm

$\small\begin{array}{|c|c|c|c|c|c|} \hline \text{dimensions} & \text{non-zero elements} & \text{f1} & \text{f2} & \text{f3} & \text{f4} \\ \hline \{101,101\} & 100 & 0.000322 & 0.000279 & 0.000131 & 0.00487 \\ \hline \{100000,100000\} & 100 & 0.0017 & 0.0053 & 0.004 & \text{n/a} \\ \hline \end{array}$