We can reduce the time by 3 times using the change of variables and method `"AdaptiveMonteCarlo"`. In this case, the integral is calculated without messages and errors, we have code

 \[Epsilon] = 10; y = r Cos[p] Sin[t]; z = r Sin[p] Sin[t]; \[Tau] = 
 r Cos[t];
 d = x^2 + y^2 + z^2;
 I15 = 1/((1 - x)^2 + y^2 + z^2 + \[Tau]^2) // FullSimplify;
 R = (1 + \[Tau]4^2) I15;
 S = (x^2 + y^2 + z^2 + (\[Tau] - \[Tau]4)^2) I15 // FullSimplify;
 a = 1/4 Sqrt[4*R*S - (1 - R - S)^2];
 F = I Sqrt[-((1 - R - S - 4 I*a)/(1 - R - S + 4 I*a))];
 Phi = 1/a Im[
 PolyLog[2, F Sqrt[R/S]] + Log[Sqrt[R/S]]*Log[1 - F Sqrt[R/S]]];
 integrand = I15^3/d^(1/2) (4 \[Tau]^2 I15 - 1) Phi;
 In[14]:= NIntegrate[
 integrand r^2 Sin[
 t], {\[Tau]4, -\[Infinity], \[Infinity]}, {x, -\[Infinity], 
 1 - \[Epsilon]/2}, {r, 0, \[Infinity]}, {p, 0, 2 Pi}, {t, 0, Pi}, 
 Method -> "AdaptiveMonteCarlo"] // Timing
 
 (*Out[]= {217.781, -0.164111}*)