I have a 150A Square D QO panel that is full and won't allow for tandem circuits. I'm going to take out the bottom 2 breakers and replace them with a 100a breaker to feed a subpanel via 4AWG copper wire. When I move the 20 amp circuits from the main to the subpanel, do I need to move the neutral wires as well, or can I leave them on the neutral bus in the main panel? Seems like its all the same if I left them alone, but not sure if there is a reason to move them too.
You'll need to move them to the subpanel
You'll need to reroute the neutrals to the subpanel along with their associated hots (300.3(B) is the closest thing that applies); this has the advantage that the rerouted branch circuits can occupy a different conduit or cable from the feeder, though, easing derating.
And that 4AWG wire is a size too small
Your other problem is that 4AWG copper is one size too small for a 100A feeder in the general case; you'll need to either downsize to a 80 or 90A breaker or upgauge the wires to 3AWG. Note also that if your AHJ permits it, 1AWG aluminum is much more cost effective for a feeder this size.