It may or may not burn up the motor. It depends on how the motors are wired (if they can take the 6s) Maybe they can take 4s like barbers? It WILL give the starter more kick and rpm. Again the motors have to be able to take this whopping increase in power. If it's a "12 v" starter than it might be able to do 4s.

This could also be your battery pack or a connection too. Some measurements would be in order to narrow the trouble down. If it is a sealed can motor then it's probably not possible or feasible to acquire or replace the brushes. You surely can put a bigger motor in it though. If you selected a slower/bigger one then it may take advantage of the higher voltage and make a "real" starter out of it.

ps. One quick test you can do is to parallel another 3s pack temporarily to see if it's the battery's fault. Or perhaps measure the volt/current on the set-up.