I pushed my first set of F# code to Azure today. Upon first run, I got an error that FSharp.Core could not be found. I was pretty sure it was in the package that was deployed, so I set about a bit investigation. The most appropriate link I found was the following:
http://code.msdn.microsoft.com/fsharpazure
Basically, you can run F# code in azure, but because F# libraries are stored locally in the GAC (and hence referenced in the GAC), and that since in Azure the the GAC F# assemblies to not allow partial trust the F# libraries are not found by default?
The solution? It's in that article to a point. My F# code was not an actual worker_role piece of code, so that template did not assist me. My worker was a C# worker that was attempting to call an F# library. Since I had the code, I put the --standalone value in the other flags section of the project properties (see the picture), recompiled, linked the new dll into the C# app, deployed it, and it ran fine!