If two players both want to pass priority, does the spell or ability on the stack resolve?

When both players pass priority consecutively without either player taking any action, the game moves forward. If there is a spell or ability on top of the stack, it resolves (CR 117.4). The key condition is that all players must pass priority in succession with nothing new being put on the stack in between.

In a two-player game, the active player receives priority first after a spell or ability is placed on the stack. If the active player passes, the non-active player then receives priority. If that player also passes without acting, the top object on the stack resolves (CR 116.4). After resolution, the active player receives priority again.

If the stack is empty and both players pass priority in succession, the current step or phase ends instead — no spell or ability resolves because there is nothing to resolve (CR 116.4, CR 500.4).

Example: You cast Lightning Bolt targeting your opponent's creature. Your opponent passes priority without casting any spells or activating any abilities, and you also pass priority. Since both players passed in succession, Lightning Bolt resolves and deals 3 damage to the creature.

In multiplayer games, all players must pass priority in succession (going in turn order) for the top object to resolve, not just two of them (CR 117.4).