This turned out to be an issue with different versions of the F# compiler being used. Using Visual Studio 2013 RC, I was using the 3.1 version, which has had some improvements causing my specific issues to not appear.
The problem was made difficult to identify because changing my F# project to use the 3.0 compiler in Visual Studio did not have any effect (it still compiled just fine). I even opened the solution up in Visual Studio 2012, and it all compiled well enough. It was only when I created a new project from scratch in VS 2012 that I began to get the same compilation errors as on the server. So I guess that this must be an error (or at least not very desirable feature) in the project file that was created. I'm not sure if this should be classified as a bug with VS 2013, especially since I don't know exactly what causes the error.
So the simple solution until Azure (or is this a Kudu thing?) supports the 3.1 compiler is to either recreate the project file in VS 2012 or to copy code which does not compile over to a test solution where fixes can be tested.