If I tap a bunch of land and then it gets destroyed after I tapped it, does the mana stay in my mana pool?
Yes, it will stay in your mana pool until the end of the current step or phase.
There are effects that empty your mana pool, but they specifically say so, such as Mana Short.
106.4. When an effect produces mana, that mana goes into a player’s mana pool. From there, it can be used to pay costs immediately, or it can stay in the player’s mana pool. Each player’s mana pool empties at the end of each step and phase.
Once mana has been added to your mana pool it will remain there until the end of each step unless you choose to spend it to pay a cost.
The land is tapped for mana and that mana is available until the end of the current phase or step.
For example, the Combat phase has several steps.
The combat phase is the third phase in a turn, and has five steps in this >order:
Beginning of Combat Step
Declare Attackers Step
Mana tapped, then the land being destroyed sounds unlikely to happen all in a single phase or step, as land destruction is often sorcery speed. You would have to be tapping on opponent's turn, or they have instant speed destruction, but yes, the mana stays until end of the step, within the phase.
I can go into a really long, technical explanation of how/why but the short answer is yes. Yes, the Mana stays in your mana pool until the end of the step/phase the lands are tapped in.