Remove Account Without Resetting Phone I have a Galaxy Nexus. However, I know someone that is having the exact same issue that has an older Android phone. 

The account I want to remove is not the Google account. We have Lotus Notes Traveler installed. Recently there was an update and that's when the problem started. I am unable to uninstall the app because of the account it uses. When I try to remove the account it says that I cannot remove it without factory resetting the phone. 

Neither one of our phones are rooted. 

Is there some way to remove the account without resetting the phone?
Short answer: No.

The app dev must have made it like this. You could email the dev, to ask them to sort it, but otherwise there is no option.