What happens to the "leftover" Ions in the Mass Analyzer
Most would end up being sucked out though the vacuum pump with the other residual gases. I assume that a lot of them collide with the sides of the vacuum chamber and are neutralized before sublimating in the vacuum.

The quads can become contaminated and require cleaning after several years of heavy use. Because there are such small quantities at that stage of the process, it's not uncommon for an MS to reach the end of it's service life before the quads need to be cleaned.