I purchased applications from multiple iTunes store accounts, and when I sync with iTunes they are removed from the device. What do I do in this situation?
Likely this means you are not signed in and authorized with the secondary account you purchased the applications with, generally iTunes will report something on the lines of "You are not authorized to playback this content" blah blah I am going to erase all the stuff you are not authorized to play.
The fix is to go to the Store menu in iTunes and choose authorize computer, and enter in the secondary credentials, then you should be able to sync purchased apps from multiple account holders using the transfer purchases option, if nothing else.
Apple has some decent articles on the process:
About iTunes Store authorization and deauthorization
iTunes Store: Transferring purchases from your iPhone, iPad, or iPod to a computer