Does a creature die immediately when its toughness drops to 0 from a continuous effect?

Whenever a creature's toughness is reduced to 0 or less by a continuous effect (such as a static ability, an Aura, or an Equipment), state-based actions check for this condition and the creature is put into its owner's graveyard. This happens before any player receives priority to act. The relevant rule is CR 704.5f: "If a creature has toughness 0 or less, it's put into its owner's graveyard."

State-based actions are checked continuously — specifically, they are checked every time a player would receive priority (CR 704.3). This means no player can respond to the creature 'being at 0 toughness'; the creature simply ceases to exist on the battlefield the moment the game checks state-based actions after the effect is applied.

This is different from lethal damage (CR 704.5g), which requires that damage equal to or greater than the creature's toughness has been marked on it. Reducing toughness via a continuous effect is its own separate state-based action check under 704.5f, so both can apply simultaneously — for example if toughness is reduced AND damage is marked.

Concrete example: Your opponent controls a 2/2 creature. You cast Tragic Slip giving it -1/-1 until end of turn. The creature becomes a 1/1. Then you enchant it with a hypothetical Aura that gives -1/-1 continuously — the creature drops to 0/0. Immediately when state-based actions are checked, before anyone can cast a spell or activate an ability, the creature is put into the graveyard due to CR 704.5f.

Note that if a spell or ability would temporarily reduce toughness to 0 and then immediately restore it (all within one effect), state-based actions haven't been checked in between, so the creature may survive — but once SBAs are checked and toughness is still 0 or less, the creature dies.