Suspend is a keyword ability that lets you cast a spell "for its suspend cost" — but this is actually an activated ability you activate rather than a normal cast. According to CR 702.61a, suspend's activated ability can only be activated when you could cast the card normally. For sorceries and most permanents, that means during your own main phase when the stack is empty (CR 307.1). You cannot activate suspend during your opponent's turn for those card types.
When you exile the card with suspend counters, it waits in exile. At the beginning of each of your upkeeps, you remove one time counter (CR 702.61b). When the last counter is removed, you cast the card without paying its mana cost — this can happen at instant speed since it's triggered during an upkeep step, but the original activation must have been legal.
If the card with suspend happens to be an instant (or has flash), then you could activate suspend during an opponent's turn, since instants can be cast at any time you have priority (CR 702.61a, 116.2b). So the answer depends on card type.
Example: You want to exile Ancestral Vision (a sorcery with suspend) during your opponent's turn. You cannot — the suspend ability may only be activated during your own main phase when the stack is empty. You must wait until your turn.
In summary: for sorceries and permanents, suspend cannot be activated on an opponent's turn. For instants or cards with flash, it can be activated whenever you have priority.
Unofficial fan resource — not affiliated with or endorsed by Wizards of the Coast. Answers are AI-generated estimates grounded in the Comprehensive Rules and are not a substitute for an official judge. Verify anything match-critical.