I bought 2 Hunter Exeter model fans and was running into the problem of the lights cutting in and out after about 10-15 minutes of steady light output. Both fans were having the same problem. The fans are controlled by radio frequency remote controls - which the instructions say are "paired" with the fan when they arrive. BUT, you can re-pair them and doing so solved my intermittent light problem. I only re-paired one of the fans and now both fan lights work properly. My guess is there was some interference between the remotes for the two fans, and re-pairing one eliminated that interference.
Is the fan safe to operate?
A burning smell from an electrical appliance (other than a toaster) is a sign that it is time to replace the item.
Sizzling sounds combined with burning smells are sometimes a precursor to burning the house down.
Melted electrical insulation reduces the electrical safety of the item and could lead to electric shocks (and possibly death).
I'd stop using it until it is removed (with power off at circuit breaker) and repaired or replaced.
What should I check for?
I'm not familiar with these units but, from your description, it is probably a fault in a switch operated by the chain. It may be that dirty contacts at one position are causing arcing. Arcing causes a sizzling sound, Arcing causes heating and that causes barning smells as nearby non-metal parts are burned (eventually they may melt or catch fire)
Arcing is when sparks are jumping across a small air gap where the switch contacts are pitted, corroded or dirty.
In principle it may be possible to clean-up the contacts (which may only be a temporary cure) or replace the switch (if spare parts are available). You might need to replace any other parts damaged by heat.
Best Answer
It's probably an optical illusion. The "wagon wheel effect"
You are probably viewing it in light produced by a fluorescent lamp rather than an incandescent lamp.
There do exist electric motors that can work in either direction but no motor will speed up then stop and start working in the reverse direction all by itself.