Ok, thanks for the work and information.
The remote OFF button is not providing the contact and signal to the receiver unit at the fan.
There is a receiver that accepts frequency signals from the remote control.
It has lost the ability only on the Off button.
A remote that matches yours can be purchased if found, otherwise a new set of remote and receiver would need to be installed.
They are matched and work together on the same frequency to operate the fan.
I believe your receiver is in the canopy section where the fan attach's to the ceiling and not in the fan housing itself.
A picture of the remote would help, or CLICK HERE
and see if you see one like yours.
Without a Model number may be hard to find just a new remote.
That same company in the link, has the receiver and remotes together or you can get them at any Building supply like Home Depot, Lowes, Sears etc.
A set is usually the easiest, nut maybe you will see yours on the site.
It can be ordered directly from them at the service phone number.