Alexey Popkov and Szabolcs Horvát, thanks for answers.

In fact, I do not need a symbolic pre-evaluations of the integral, so this method works well for me.

This error message comes from symbolic pre-evaluation performed by the outer NIntegrate and has nothing related to Block. The best is to combine the two NIntegrate into one:

y = 1;
NIntegrate[E^(-x y z), {x, 0, 1}, {z, 0, 1}]

(* => 0.7965995998164899`  *)

If the latter isn't possible, a workaround is to disable symbolic preprocessing by turning the inner integral into a black-box function (but it can considerably slow down integration):

Clear[f]
f[x_Real, y_] := NIntegrate[E^(-x y z), {z, 0, 1}]
Block[{y = 1}, NIntegrate[f[x, y], {x, 0, y}]]

(* => 0.7965995992970549`  *)