The performance of a migrated .NET application, when compared to the original VB6 application, is normally very similar or better. The only instances when we have seen a performance decrease is when doing an important re-architecture or when the database engine is changed (from Access to Oracle, for example). Every once in a while we also run into issues with the Garbage Collector, but fortunately they are not that common and are easy to detect.

One thing to keep in mind is the way .NET loads assemblies, and how they are executed. .NET assemblies are compiled to an intermediate language, called CIL (Common Intermediate Language, formerly MSIL). When these assemblies are executed, the default behavior for the .NET Framework is to use a Just-In-Time compiler, which compiles the CIL code for a method to native code "on the fly" the first time the method is called. This implies an overhead on this first call, which suffers from a (normally acceptable) performance impact while the JIT compiler runs. Once a method is compiled, though, it is kept in memory, so the performance of subsequent calls is not affected. Once the code is in memory the performance of the .NET application is normally better than the performance of the VB6 application. You can find more information on JIT here.

It is worth mentioning that you can run ngen.exe on the application to compile the CIL assemblies to native code. This may improve application startup and first run times. Here's an interesting take on whether to ngen or not to ngen.

Check out this performance optimization book in MSDN, called Improving .NET Application Performance and Scalability. It is slightly outdated, but the concepts still apply and are useful to improve the performance of migrated applications.