This gives a result:
eqn = D[u[x, y, z, t], t] ==
D[u[x, y, z, t], x, x] + D[u[x, y, z, t], y, y] +
D[u[x, y, z, t], z, z];
cond = {u[x, y, z, 0] == (1 - y - z) E^x,
u[0, y, z, t] == (1 - y - z) E^t ,
u[1, y, z, t] == (1 - y - z) E^(1 + t),
u[x, 0, z, t] == (1 - z) E^(x + t),
u[x, 1, z, t] == -z E^(x + t),
u[x, y, 0, t] == (1 - y) E^(x + t),
u[x, y, 1, t] == -y E^(x + t)};
NDSolveValue[{eqn, cond}, u,
{x, 0, 1}, {y, 0, 1}, {z, 0, 1}, {t, 0, 1}]
