Hello,
I am not sure what the purpose of "f[x, y] == oper[Sin[x] Sin[y]];" is for, but this will work:
oper = (D[#1, x, x] + D[#1, y, y]) &
Plot3D[Evaluate[oper[Sin[x] Sin[y]]], {x, 0, 10}, {y, 0, 10}]
Plot and related functions have the property of HoldFirst---the upshot is that you need to Evaluate explicitly.
I think if you google Mathematica and Plot and Evaluate, you will see some excellent discussions of this.