<p>How do I determine the dependencies of a .NET application? Does <a href="http://www.dependencywalker.com/" rel="noreferrer">Dependency Walker</a> work with managed apps? I've downloaded the latest and tried profiling the app, but it just exits without much of an explanation. If it doesn't work with .NET, then is there some other tool that would help me debug a run-time DLL loading issue?</p>

Dependency walker works on normal win32 binaries. All .NET dll's and exe's have a small stub header part which makes them look like normal binaries, but all it basically says is "load the CLR" - so that's all that dependency walker will tell you.
To see which things your .NET app actually relies on, you can use the tremendously excellent
.NET reflector
from Red Gate. (EDIT: Note that .NET Reflector is now a paid product.
ILSpy
is free and open source and very similar.)
Load your DLL into it, right click, and chose 'Analyze' - you'll then see a "Depends On" item which will show you all the other dll's (and methods inside those dll's) that it needs.
It can sometimes get trickier though, in that your app depends on X dll, and X dll is present, but for whatever reason can't be loaded or located at runtime.
To troubleshoot those kinds of issues, Microsoft have an
Assembly Binding Log Viewer
which can show you what's going on
at runtime