I had the problem when I first got the phone. Switched over to another free app called Mail Droid and the problem stopped. However, I went back to the Gmail app due to some issues with Mail Droid, and it was fine for about 2 weeks. Now the problem seems to be creeping back again where e-mails don't come in unless I manually refresh. Same goes with Google Voice text messages.
I have an additional problem with both programs when it comes to being connected to wi-fi though. E-mails and Google Voice text messages come in whenever they want, and typically that is no where near instantaneously. Yesterday, I received a Google Voice message at 7 pm. It was sent at 2 pm.