Suave is a possible replacement for the ASP.NET stack, but with some help, the two can be used together to build composable F# web applications.
In my opinion, Azure App Services are an easy and direct way to launch your ASP.NET Core application for the outside world to access. Azure App Service offers a free tier, perfect for tinkering or getting a new project off the ground.
While not the primary focus of Microsoft's efforts, F# makes for an excellent language choice for applications targeting .NET Core.
Linux and Microsoft's .NET framework haven't always seen eye to eye. The Mono Project looks to get Linux and .NET on the same page.