Replace batteries, they can shows good voltage but on load, voltage rapidly go down, due to some internal chem material failure. In this situation batteries cant supply needed current to circuit.
Clean IR diode body or clean protecting glass/plastic for IR diode.
this is most likely if the capacitor inside your fan has gone bad. Ceiling fans are usually single split-phase induction motors which require a capacitor to work. Probably nothing to do with the remote. However since you have a speed control based on electronics, then probably there are SCR's/Thyristors involved, which can make fan capacitors go bad early.