After being rescued, he will wait in the room with the blue door. When Ronald Shiner complains about the lack of food, he rallies up the other survivors in the same room, including Burt, to head out and look for food.
If Frank does not stop Ronald's mutiny, Burt will leave the Security Room along with the other survivors in the same room. They will no longer count towards the number of rescued survivors.
Burt's picture is used for the Item Smasherachievement. This could be because he attempts to hit Frank with a baseball bat when they first meet.
If Aaron dies before Burt is recruited, he will absolutely refuse to follow Frank; accusing him of Aaron's death.
On the other hand, if Frank kills Burt, Aaron will scream "AH!!! No, get away from me! Don't kill me!". Interacting with him again, Frank will say "Unless you're in a hurry to die, you shut the hell up an follow me! We don't need any extra noise." Aaron joins after this.
Aaron and Burt's barricade will remain intact if Frank tears it down without recruiting either of them and exits the plaza before entering again.