Can you use a land's activated ability to produce mana during an opponent's turn without any spell or ability on the stack?

Mana abilities are a special category of activated abilities that produce mana as part of their effect. Under CR 605.3a, a player may activate a mana ability whenever they have priority, or whenever a rule or effect says they may add mana. This is a broad permission that is not restricted to your own turn or to times when something is on the stack.

However, simply having mana in your mana pool does not let you cast spells or activate abilities outside of the normal timing rules. The mana will be produced and sit in your mana pool, but unless you have a reason to spend it (such as responding to a triggered ability, or casting an instant), it will be lost when the phase or step ends due to mana burn being replaced by simply emptying the pool (CR 106.4).

There is rarely a practical reason to tap a land for mana on an opponent's turn with nothing on the stack, because you cannot spend it on a sorcery-speed spell or ability, and unused mana empties from your pool as steps and phases end. The exception is if you plan to activate an instant-speed ability or cast an instant in response to something about to happen.

Example: It is your opponent's main phase and they have not cast anything. You tap a Forest for one green mana. The mana enters your mana pool legally, but since you have nothing to spend it on at this moment, it will drain away at the end of the main phase step. If instead your opponent then casts a spell, you would still have priority to spend that mana in response.

Key rules: CR 605.3a (when mana abilities may be activated), CR 605.1 (definition of mana ability), CR 106.4 (mana empties from pool).