NContourIntegrate[f_, par : (z_ -> g_), {t_, a_, b_}] :=
  NIntegrate[Evaluate[D[g, t] (f /. par) /. t -> t1], {t1, a, b}]

Clear[Pinfz1];
Pinfz1[p_?NumericQ, z2_?NumericQ] :=
  NContourIntegrate[1/(2*I*Pi)^2*z1*z2/(p + (1 - p)*z1*z2-z2)*E^(1/z1 + z1 + 1/z2 + z2),
   z1 -> Exp[I t],
   {t, 0, 2*Pi}]

Clear[Pinfz];
Pinfz[p_?NumericQ] := NContourIntegrate[Pinfz1[p, z2], z2 -> Exp[I t], {t, 0, 2*Pi }]

ii = z1*z2/(p/z2 + (1 - p) z1 - 1)*Exp[1/z1 + z1 + 1/z2 + z2];
i1 = (ii /. {z1 -> Exp[I*t1], z2 -> Exp[I*t2]})*I*Exp[I*t1]*I*
   Exp[I*t2];
f[pval_] :=
 Quiet[NIntegrate[
   Evaluate[i1 /. p -> pval], {t1, 0, 2*Pi}, {t2, 0, 2*Pi}]]
f[1/2]
f[1/3]

(* Out[114]= 89.3519154317 - 5.40968238685*10^-8 I

Out[115]= 62.3532282953 + 0.000124769719229 I *)

f[.1]

(* Out[116]= 34.4365223939 + 0.000414467508089 I *)

f[.9]

(* Out[117]= 170.924622109 + 0.00264992865898 I *)