x DiracDelta[x] // Simplify
(* 0 *)

test = ( I + ζ DiracDelta[ζ])/  ζ;
InverseFourierTransform[#@test, ζ, z] & /@ {Identity, Simplify}
(* {(1 + π Sign[z])/Sqrt[2 π], Sqrt[π/2] Sign[z]} *)

test=(I+(ζ-ζ1) DiracDelta[ζ-ζ1])/(ζ-ζ1);
InverseFourierTransform[#@test,ζ,z]&/@{Identity,Simplify}

SameQ@@%

test = (I + (ζ - ζ1) DiracDelta[ζ - ζ1])/(ζ - ζ1);
(InverseFourierTransform[#@test, ζ, z] & /@ {Identity, Simplify}) /. ζ1 -> 0

test = (I + ζ DiracDelta[ζ])/ζ;
InverseFourierTransform[#[test /. ζ -> (ζ - ζ1)] /. ζ1 -> 0,
 ζ, z] & /@ {Identity, Simplify}