Just talking from previous experience, the only device I've seen handle 'push' well is the BlackBerry. I've used Windows Mobile, Palm, and my iPod Touch and whenever you set push, it will negatively affect the battery life significantly due to the constant data connection to the server. This should be the same result with Zimbra or Exchange since they use the same protocol.
After a day of 'push' on my iPod Touch, I turned it back to polling the server every 15-30 minutes.
DJS