One thing that can cause this error is running the web role itself, instead of running the containing cloud project. If this is the issue, you could fix it by ensuring that the cloud project is set as the startup project for debugging, and not the web role.
It's possible, and sometimes useful, to run the ASP.NET project that defines the web role on its own. This can be a lot quicker than running things in the Azure Compute Emulator. It may also enable you to develop your project without having to run VS elevated. Also, I've found that the emulator tends to cause Visual Studio to report an invalid memory access error from time to time, at which point you need to restart VS. Running the web role directly avoids all these problems.
However, there are some things that can prevent this from working, and the exception you describe is a symptom of one of these problems. If your web role's Web.config includes configuration for Azure's DiagnosticMonitorTraceListener (and Visual Studio adds that by default when you create a web role) then the first thing that tries to generate trace output will crash with the error you describe if you run outside the emulator. And as it happens, retrieving a setting from the CloudConfigurationManager appears to do this.
This isn't peculiar to the CloudConfigurationManager by the way. All it's doing is producing some trace output. VS configures web roles to send all trace output to the Azure diagnostic listener, and because that listener can only run in either the compute emulator or an actual Azure instance, the first thing that tries to produce trace output will crash. CloudConfigurationManager is a common candidate because it happens to produce trace output, and it typically gets used early on when a role starts up. But in principle, anything that produces trace output could hit this exception.
A simple way to avoid this is to remove the relevant section from the configuration file. When you create a new web role, Visual Studio adds a <system.diagnostics> section that configures the default trace output to go to the Azure diagnostic listener. You could just comment that out. That will enable you to debug the web role directly in Visual Studio without using the compute emulator (assuming you aren't doing anything else that depends on being in a role environment).
Of course, the problem with that is that you'll no longer get any diagnostic traces when running in Azure. One way to solve that is to move the relevant configuration to the Web.config.Release file (adding the necessary xdt: attributes).
This change will also stop the Azure diagnostic trace listener from running when you use the local compute emulator. (That's less of a problem, because the trace messages will still appear in the debugger. It just means you won't get persistent copies of the traces copied to table storage like you would when running for real.) The obvious way to fix this would seem to be to make a similar modification to Web.config.Debug (or to run the release build in the emulator), but there's a snag: apparently cloud projects do not apply configuration file transforms when packaging for the emulator by default. Fortunately, you can fix this: http://blog.hill-it.be/2011/03/07/no-web-config-transformation-in-local-azure/ shows how to enable transforms for local debugging in the compute emulator. (Transforms are never applied when debugging an ASP.NET project directly from within VS, by the way.)
