Does the quarter round need to be removed before sanding a hardwood floor for refinishing? If so, do you replace the quarter round? If not, do you hand sand it?

No- quarter round does not have to be taken up. If it is stained or polurethaned, it can be finished as the floor is. If it is painted, it is best to paint after the floors are finished and really dried well. My rule of thumb - no walking on floors for 48 hours between coats -- or for 4-5 days after final coat. (reason: Floors may feel "dry" are often still "tacky" under the top layer. When walked on, this layer "slides" and causes areas that need sanding and re-coating. "Tacky" floors are like a piece of tape-everything like dust, lint, debris, etc. sticks to it and have to be sanded to get them smooth again.
Cleaning after sanding and patience between coats are the key to a great floor finish.
Good Luck