Maintaining multiple build versions of asp .net / SQL Server applications
This would be an unusual deployment pattern: normally you'd want to have 1-to-1 correspondence between an application and a database. This mainly is because you'd want your development environment to mirror your future production environment as much as possible.

If you cannot create databases on the server you could still look at pre-creating a set of deployment slots (e.g. QA1-QA5) and re-using them.

One thing that might help would be something like DbUp - it allows you to easily deploy just the delta scripts that are needed.
Alternatively you could look at DB change management software like DBGhost or RedGate SQL CI.