Ceiling fan light comes on/stays on - NON remote
It sounds like you have a chafed/exposed wire that is shorting out intermittently due to vibration from the fan motor; either that, or the pull switch itself is failing intermittently.  If you can get it apart and replace the pull switch (Hunter should be happy to sell you the appropriate spare part if you have the model number of your fan handy), try that, and also inspect the wiring closely for bared/chafed spots -- suitable appliance wire for replacing damaged wires should be available at your local hardware or big-box store.