Sorry, Tweetlogix does not offer its own push notification service. There a number of reasons for this, most notably that Twitter is no longer permitting additional developers to access its Site Streams API. Site Streams is the underlying Twitter API that is needed for the development of a Twitter push notification server. Site Streams is currently in a closed beta with some Twitter developers who were invited into the beta program several years ago.
However, Tweetlogix does offer an option to enable iOS Background Refresh for infrequent local notifications of new mentions and direct messages. The major disadvantage of this option is that notifications are not instantaneous - they occur only infrequently at times determined by iOS. For details, see this FAQ explaining Background Refresh.
If you’re needing instantaneous Twitter notifications, a possible alternative is to configure Boxcar to receive Twitter push notifications and open Tweetlogix. For details, see this FAQ explaining how to configure Boxcar.