Can you tap a land for mana during an opponent's turn?

Tapping a land to produce mana is a mana ability (CR 605.1a). Mana abilities don't use the stack and can be activated whenever you could normally take an action — but more importantly, they can also be activated any time you are about to pay a cost that requires mana, even without priority (CR 605.3b).

During an opponent's turn, you still receive priority after each spell or ability resolves, and during certain steps and phases (CR 116.3b). While you have priority, you may activate mana abilities freely. This means you can tap lands to float mana in response to an opponent's spell, to pay for a triggered ability, or to cast an instant or activate an ability during their turn.

It's important to note that simply tapping a land for mana without spending that mana will cause the mana to drain from your mana pool at the end of each step or phase (CR 500.4), though there is no further penalty for this in modern Magic rules.

Concrete example: Your opponent casts Lightning Bolt targeting you on their turn. You receive priority and want to cast Counterspell. You tap two Islands for {U}{U} — this is perfectly legal — then cast Counterspell targeting the Bolt.

In summary, there is no rule restricting mana abilities to your own turn. As long as you have priority or are paying a cost, you may tap lands for mana on any player's turn (CR 605.3a, 116.3b).