The ForAll syntax to use in this case is ForAll[x,cond,expr], not ForAll[x,expr]:
Reduce[ForAll[{x, y, z},
{{x, y, z} > 0, 2^x == 3^y == 5^z},
3 y < 2 x < 5 z],
Reals]
True
The way you did it, you asked whether both expressions 2^x == 3^y == 5^z and 3 y < 2 x < 5 z are true for all positive triples, which is obviously false.
