If I play a card effect on a support card and my opponent responds with a card that destroys it like demolition, will the card effect still resolve? I believe it would because it wasn't negated and is still on the stack but my friend thinks otherwise. Can someone please clarify?

Can you give an example?

Sure, I activate the liber mortis. My friend responds with demolition. Does it negate the effect? I imagine it doesn't cause demolition doesn't negate actions, so it is still on the stack and should resolve? No?

You're right, the action is independent of the source.