Nothing happens other than him getting a bump in the polls. No presidential impeachment has ever led to a conviction, even when they were caught in tape doing the thing they were accused of... All an impeachment has ever done its made the party that conducted the impeachment perform worse at the polls because they're seen as "playing politics."
To add to the other comment Biden would need to be impeached by the House and convicted by the Senate (where Dems currently have a majority). If it were somehow successful, Harris would become the new president until what would have been the end of Biden’s term.
Basically impossible right now, but helpful to know what the process looks like.
Dont forget though who becomes the vice president. Whoever Harris chooses. Aka she can make Bernie a stand in VP until the next election. Imagine how much that would piss off the GOP.