When I try to run this circuit, even with no input to the gate, the
motor always runs and the MOSFET heats up.
With no gate input, it will float and quite possibly turn on the MOSFET - do not run a MOSFET with a floating gate. So, it's quite possible you may have damaged the MOSFET and it will act like a partial short circuit thus the motor always runs.
The next problem is your general circuit. You have wired the MOSFET as a source follower and this is no good for what you want. With an N channel MOSFET this is the best configuration: -
Picture taken from here
Note the use of a diode or snubber network connected across the motor to prevent back-emfs destroying the MOSFET. You also need to pick a MOSFET that adequately turns on with only 3.3 volts gate-source drive.