TestDriven.NET (and apparently the Resharper test runner) will automatically pick up app.config files, making the contents available to unit tests, but they do not pick up web.config files in web projects.  After some Googling, I found this post that describes the solution.  To paraphrase, just add a post-build event that copies the web.config file to the bin folder and gives it the correct name, like so:

copy /Y “$(ProjectDir)Web.config” “$(TargetDir)$(TargetFileName).config

Thanks, Sebastien!