Run, don't walk, over to Tess
Ferrandez's blog, If broken it is,
fix it you should, which has well
scripted labs dedicated to learning
how to diagnose and debug crash, hang
and memory issues with .NET code. She
has some of the best material I've
found to date to help get you started.
Commercial memory profilers such as ANTS and SciTech are excellent resources that will show what objects are in the heap, and how they are rooted. Most commercial memory profilers have the ability to load a memory 'snap' of a process (say from your production environment).
You can capture a memory 'snap' (see Snap v. Dump) using adplus.vbs or DebugDiag. Adplus is available as part of the Debugging Tools for Windows. DebugDiag will also have some rudimentary analysis (but seems to be more reliable on unmanaged code) automagically.
Monitor the Application
For an idea on what to monitor, see Improving .NET Performance and Scalability, specifically Chapter 15.
As to how to monitor, there are commercial tools available for that as well, however, every Windows machine also comes with Perfmon.exe, which can be used to record relevant performance counters.
Test the Application
For an idea on how to perform load, or stress, tests, check out the Patterns and Practices Performance Testing Guidance for Web Applications.
Debug the Application
Once you've identified you've got a problem (monitoring) and your able to reproduce the problem (testing) you can get down to debugging the problem. See the links for Tess - that information will carry you a long way.
Then rinse and repeat! :)
Good luck!
Z