I have a code written in C# I would like to use as the back-end of a site I'm building.
I would prefer not to build the site front-end in ASP.NET (which integrates nicely with C#), and to use PHP or Python instead. 
Is that reasonable? Should I re-consider using ASP.NET?
How can I achieve that?
You can do whatever you like. Personally i wouldnt use php because i dont know very much php.
But you can do it, you could expose a soap web service and there are libraries that will let php talk to it.
No one here will be able to tell you what you haven't already told us. Asp.Net will probably be easier because of how everything integrates and you can share classes etc - but that does not mean you HAVE to use it.
Both of them are fairly passive server side technologies that present html to browsers though. why do you need 2 servers?
You have to ask why you are doing it .. if you are playing and want to learn then of course you can do it just to see how it all works. But if you are on a commercial project then id suggest that you dont need both a php and a c# server ... or if you do perhaps you want to go asp.net for your web server and if you need another layer of services behind then use WCF if you want to go a microsoft route. Howver it is usually possible to host all services in the same IIs instance.
Connecting C# Back-end with PHP frontend