I just create a new MVC 4 Web API project, and create a new .cshtml file, containing very simple HTML:
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If you have enabled Web Pages and you have verified that the .NET version is valid then your problem is most likely that the IIS server does not have the razor binaries available to be shared across all sites. To fix it you should ensure you've included the mvc binaries in your /bin folder. This link explains it very well:
http://www.codeproject.com/Answers/437259/his-type-of-page-is-not-served-CSHTML-how-to-serve#answer1