There is no automatic means to do that. As far as I know, BandsInTown does not provide an API (a way for programs to talk with each other), and only provides for gig imports. I would appreciate it if you would contact BandsInTown technical support and ask them if they have that capability. If they do, I'm happy to put automatic updates into WheresTheGig.
(This next part is just me brainstorming about a cool way to do this, and can be skipped if you're not interested)
I think a really efficient way to do a basic automatic update would be for them to take an ICal feed URL and poll it every hour or two. They could extract the gig information from the ICal feed and update the BandsInTown information. However, it might be difficult to extract some of their specialized fields, but this could be overcome using specifically formatted gig descriptions.
For those not familiar with the BandsInTown export, here's the process. First go to Exports => For Import To Other Sites:
Then select the BandsInTown option:
