Can I connect the grounding conductor to the neutral when installing a remote for a ceiling fan in an ungrounded box?
NO!
You cannot, for any reason, connect a grounding conductor to a neutral (grounded) conductor anywhere other than in the service equipment.
If you do so, the metal box and any metal connected to the box (including the fan housing) will become a current carrying conductor. This is very bad, and can result in personal injury and/or death.
I doubt the remote requires the grounding conductor to be connected in order to function.
The device's grounding conductor is a safety feature, and should be left unconnected when installing the device in an ungrounded box where there is no separate grounding wire in the feed cable.  However, you'll want to be sure the box is not grounded. Just because there's not a grounding wire in the box, does not mean the box is not grounded. The box could be grounded via conduit, the outer sheath of the cable feeding the box, or some other method.