You can build mainline linux without an MMU, and there are even pretty crazy setups where you can run it on a ARM cortex (though usually an M4). It is not a standard system though, very little software will run without modification. The biggest issue for such processors is usually actually lack of memory (they have relatively little built-in and most have no external memory busses. There's at least one project where the externel memory is bitbanged through gpio!).