It's never really been expanded upon. Turning them to stone or wood (like the mermaid) presumably just traps them, definitely so for being turned to tin (until the Tin Man was smashed to pieces somehow, rest in Olympus), but what about being turned into regular inanimate objects?
Gaston is an interesting case for this - he was turned into a rose, and shows up in the underworld, saying Rumpel killed him. While his death is definite, it is never made clear whether the cause of death was being turned into a rose in itself, or having his stem cut by Belle afterwards.
The other notable cases that I remember are Gepetto's parents being turned into puppets, Achmed and Jafar being turned into staffs, and Gothel being turned into a tree.
I know the show doesn't seem to make it 100% clear, but does turning them into these objects actually kill them or does it just trap them? (They don't seem to be able to reverse this)