omit a pairing of numbers from two data ranges in a table

another option is to implement your own region function like for listplot:

(*constraint here*)
f[x_, y_] /; (x^2 + y^2 != 0) := {x/(x^2 + y^2), y/(x^2 + y^2)}

f[x_, y_] := Nothing;  (*what to return for bad region, or other point*)

ListPlot[Table[f[x, y], {x, -10, 10, 1}, {y, -10, 10, 1}]]

if you dont mind turning off the error messages you can use

Quiet@ListPlot[Table[{x/(x^2+y^2),y/(x^2+y^2)}, {x, -10, 10, 1}, {y, -10, 10, 1}]]

but that doesn't exclude the point (0,0), it just avoid the warnings.