If the fan works poorly rotates slowly or not at all and emanates a burning smell the problem is likely in the motor or the bearings. From the second law of thermodynamics we know where that heat is going into the cooler room.
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Ceiling fan motor hot. And in the winter a ceiling fan can be reversed to circulate rising warm air. Because it stirs up a breeze in a room you can set the rooms thermostat 5 to 7 degrees higher dramatically cutting your air conditioning bills. Fans have 1 2 3 caps or a single plastic box with 2 or 3 caps inside it. So there is no way a hot motor can start a fire. The circuit breaker wont trip and the smell will persist. If you conect the black to the ground that is wrong and it will get hot.
This might be why the motor is getting hot. Replacing a ceiling fan capacitor. Look for the mf number s. If it is smoking and flaming that would be a cause for concern. If you bought the 2999 special you can use it to fry eggs. Just installed a new ceiling fanthe fan works fine but after it runs for a whilewe are smelling a paint burning smell.
The infrared image below shows a ceiling fan motor thats hotter than the room its in. Internal parts are typically not user serviceable so a replacement motor is your best bet. A hot motor means the internal bearings and components are damaged. As long as you conect the wihte to white black to black and ground to the ground that is the right way to wire the fan. I know electric motors can get very hot but this is not a compressor motor its a 60 watt fan which by definition should have good airflow over it. If any of the speeds do not start and spin then a capacitor is bad.
The motor is not supposed to be in contact with anything flammable. The fan is flush mount so the motor is encased and it feels rather hot. A ceiling fan makes a room more comfortable at higher temperatures during the summer. And the cheaper the fan the hotter it will get. Replace the ceiling fan motor if it feels hot during this test. Because electric motors are devices that turn electrical energy into mechanical energy most of which ends up as heat.
It is not normally installed directly against the ceiling and the ceiling is supposed to be non flammable drywall. You gotta figure it should be at least 50 efficient and 30 of those watts are moving air rather than heating up the housing. It will probably do that better than move air.