Sorry for the bump, but I've also wanted this feature but I found something with hosting GM apps on a Windows virtual environment. I hope it helps someone.
I managed to get GM games to run a virtual Windows environment, however you need to make sure the VPS can support 3d acceleration, or your VMWare instance has it enabled. This feature will simulate a GPU through the server's CPU, however 1 GM instance can eat your entire CPU if it has a display. I resolved this by adding a "minimalism" server mode to my game, where the game will run in a room with a 1x1 room size, and undraw absolutely everything. My game with 16 player support maybe peaked at 5% CPU usage when it was the most demanding. Naturally, the downside to this is you can't see the game but I got around this with logging and implementing RCON features if you need to keep an eye on things.
So while I have found something that works for me and I don't really have a need for the feature as much anymore, I still don't understand why some people are so openly hostile towards this adding this feature. Isn't Game Maker designed to be a tool for convenience? Headless support would make multiplayer games very convenient for developers. It's simple as that to me, efficiency or laziness doesn't really factor in because if you cared deeply about those traits you wouldn't be here. I highly doubt this feature being added will stop you from writing something yourself. Maybe YYG could charge for an headless export? I'd buy it day one, and I'm sure others would too.