This has been debated multiple times. At one time UA in the delay email message has stated you still must be available at the orignal time. While it is rare, UA has been known to pull up departure times that had been delay and there are the posts of those that were caught unaware -- either in an airport lounge or not even at the airport.

I was at IAH recently on a connecting with a delay departure and fortunately the new push feature on boarding hit my phone, a full hour before the delayed departure. Made it to the gate as they were doing BG5. We departed 20minutes earlier than the orignal delay time.

Probably most of the time you will be safe but if a FAA slot opens or if an alternative aircraft becomes available or ... UA is going to get the flight out with less delay if it can.

Hang out at the lounge but watch for departure status changes.