I can find a lot of obsolete threads but a lot have been changed the last months.
Is there currently any possibility to run .NET Core and/or ASP.NET Core on Raspberr
.NET Core is scheduled to run on ARM based OS like Raspian / Windows 10 IoT for Q4 2016 / Q1 2017. https://blogs.msdn.microsoft.com/dotnet/2016/07/15/net-core-roadmap/
At this point of time I think you can follow the installations for Debian but you should get a exec format error
IIRC.
Edit 3/10/17: first official instruction is online now: https://github.com/dotnet/core/blob/master/samples/RaspberryPiInstructions.md