MacBook Pro shuts itself off after a period of time

My MacBook Pro turns itself off after a period of time. The longer I leave it turned off, the longer it will power on for the next time I turn it on. For example, if I leave it off for about 24 hours, I'll get about 20 minutes out of it before it turns off. If I try turning it on immediately after it has turned itself off, it will get part way through the start up screen and then the screen will go black again. Sometimes it will get so bad that my HDD will spin up and then it will get in to a loop where it restarts over and over and never gets to the start screen.

When I say 'turning itself off', what actually happens is the screen will go black but the fans will stay running and the keyboard LEDs will stay turned on.

I've ruled out the battery (it exhibits the same behaviour without the battery) and the ram (I've removed and tested each module).

Any idea what this could be? I'm guessing that it's a logic board issue?

Edit: turns out that my screen turns off but my computer is still actually running (I can still fumble my way through login and if I open a terminal through spotlight, I can make it say something using the "say" command. I have also noticed that it won't sleep if I shut the lid.

Thanks in advance!