Kamala Harris would honestly be a terrific pick. She's a very bright and qualified person who's on the right side of almost every issue, but seems to have horrible political instincts and no real electoral constituency. She's a little only the old side, but just barely. And this would allow Biden to nominate a new VP with actual appeal to Latinos.
Dems would immediately lose their senate majority. It requires a majority of both chambers to select a new VP after the vacancy, and there would be no one to break the tie in the senate. Republicans would have a say in who the new VP is and they could even hold the VP seat open until 2025 if they wanted to.