Can a player tap lands for mana, in response to Elder Deep-Fiend tapping those lands at the beginning of the turn?
When you cast your creature, its triggered ability goes on the stack. At this point, all players have a chance to respond to the ability, before it resolves and taps the lands. They can respond by tapping the land for mana. The mana that is produced will go away at the end of the upkeep step, however, so they would have to use it then; they can’t keep it to use later.

So you can use this to tap their lands, but they have a chance to respond with an instant effect during the upkeep. It will prevent them from using those lands during their main phase.