Simply processing the pages through an ASP application server isn't the only thing that you'll need to consider. Are your sites connected to a database? If so, which db engine?
If using Access I am pretty sure that you can't access the default sql engine functionality from Linux. I believe Linux is capable of working with Windows dlls that would allow this but I doubt they're very well supported, and it's unlikely that many people would provide a decent port to M$ from Linux considering the significant divide between the two camps.
If you're using MSSQL there is no support for that here. You'd have to point to a database server elsewhere. Once you go that far, you probably might as well run the files at that same location.
Maybe I'm off base here, these are just some thoughts that I had on the subject.