With the fixes I did in the snapshot released on October 29, it is now possible to run Minecraft on IKVM.NET (on Windows). To be clear, I'm talking about the Minecraft client, not the server that has been running on IKVM.NET for a long time.
The Minecraft client uses native code for the 3D graphics and sound, so it doesn't run into IKVM limitations there.
To get it to run download the most recent IKVM.NET snapshot (currently available here) and unzip it. Download minecraft.jar and run it like this:
ikvm -jar minecraft.jar
It takes a while to start up, so be patient.
If you get an exception when trying to log in, you may have to visit https://authserver.mojang.com/ in Internet Explorer to add the root certificate to the Windows certificate store (just visiting the site causes IE to download it from Microsoft). After that you have to restart Minecraft.