An unregulated supply will be fine if you're just driving the bed, though you may see small weird fluctuations in temp sometimes (depends how much load throws out the voltage).
I wouldn't run motors or the hot end on it though.
As to what will happen:
- PSU cuts out once over 10A (current limited PSU)
- PSU internal fuse blows (will require opening up, which if not open-able means your PSU is dead)
- PSU internally cooks some component (dead and possibly not worth fixing, even if you can open it)
- PSU has enough internal lee-way and you'll get 11-12A out of the 10A rail (voltages may be unstable though and you will see "ringing" on the rails - induced AC noise or unfiltered ripple)