Sorry, Tweetlogix does not offer its own push notification service. However, there is an option to enable iOS background refresh for infrequent local notifications. Another possible alternative is to configure Boxcar to receive push notifications and open Tweetlogix.
Background Refresh is a feature of iOS which enables apps to occasionally awaken from a suspended state to refresh themselves. In Tweetlogix, you can enable background refresh for Twitter Mentions and Direct Messages.