ClearAll[FiniteDifference];
FiniteDifference[expr_, xs_List, ds_List] := Block[{},
   If[Length[xs] == 0,
    expr,
    If[Length[xs] == 1,
     If[
      ds[[1]] == 0,
      expr,
      If[ds[[1]] == 1,
       (expr /. xs[[1]] -> xs[[1]] + 1) - expr,
       FiniteDifference[(expr /. xs[[1]] -> xs[[1]] + 1) -
         expr, {xs[[1]]}, {ds[[1]] - 1}]
       ]
      ],
     FiniteDifference[FiniteDifference[expr, {xs[[1]]}, {ds[[1]]}],
      Rest[xs], Rest[ds]]
     ]
    ]
   ];
FiniteDifference[f[x], {x}, {0}]
FiniteDifference[f[x], {x}, {1}]
FiniteDifference[f[x], {x}, {2}]
FiniteDifference[f[x, y], {x, y}, {0, 1}]
FiniteDifference[f[x, y], {x, y}, {1, 1}]

Out[123]= f[x]

Out[124]= -f[x] + f[1 + x]

Out[125]= f[x] - 2 f[1 + x] + f[2 + x]

Out[126]= -f[x, y] + f[x, 1 + y]

Out[127]= f[x, y] - f[x, 1 + y] - f[1 + x, y] + f[1 + x, 1 + y]