Kirby's Stone could have been changed in Brawl. But let's assume it is the same move for argument's sake. Many Final Smashes are, I believe, considered "Grab" hitboxes such as Ike, Link, Meta Knight, etc. If this is true, that would bypass Kirby's protection while in Stone.
There are also Finals Smashes which do not appear to be "Grabs" but instead act more like projectiles. Obvious ones being Mario, Samus, etc. but also not so obvious ones such as Pit's. You can even say Pikachu, Sonic, etc fall into this category since they make themselves into powerful, unstoppable projectiles. These types of Final Smashes will most likely be resisted by Kirby's Stone, but you have to remember that Stone is not totally invunerable like it is in Kirby's video games. It can only take a small amount of damage before "breaking" (about 32%), which is not difficult to deal with Final Smashes.
So, I think it should be obvious what will happen if you try to use Stone to avoid Final Smashes. You'll either get "Grabbed" by them, or take so much damage the Stone "breaks". But if there's a Final Smash that requires a weak, initial hit to be delivered before it is executed (Link's?) then Stone could prevent that damage and the ensuing Final Smash in theory.
