When do resource-generating skills work on destructible objects?

It depends on the class:

• Barbarians generate fury whenever they successfully hit an enemy or destroyable object with a fury generator.

• Monks generate spirit whenever they successfully hit an enemy (but not destroyable object) with a spirit generator.

• Demon Hunters regenerate hatred merely by using a hatred generating skill, whether it hits or not.

The other two classes don't have "generator" skills and rely on passive generation.

According to Blizzard (under "Combat"):

Due to degenerative nature of the Barbarian's Fury, it should be generated by attacking environment objects, but since Monk's Spirit is not degenerative, it should not be generated by attacking environment objects.