Any general reasonable performance tests for DBs on application server and DBs on separate server?
If you can capture a typical workload with Profiler, you could replay it against a point-in-time restored copy of the database via the RML tools. Summary of the RML tools can be found in Testing Stored Procedure Scalability.

What you decide to measure as part of these tests in dependent on the application. Measuring response time is simple for a web application for example, not so easy with a desktop client.

It would be rare to find that performance degrades when you move the database to a dedicated server. Typically this only occurs when the application is overly "chatty" i.e. executing lots of iterative queries (incorrect use of an ORM a common cause), rather than set operations. In this situation the additional overhead of marshalling data over the wire via TCP or Named Pipes vs Shared Memory Protocol outweighs the performance gained from dedicating a server to SQL.