What do I do with the unused wire for ceiling fan installation?
The red wire is the ungrounded (hot) conductor from one switch, while the black wire is the ungrounded (hot) conductor from the other switch.  In the original installation one switch would energize the red wire causing the fan to come on, while the other would energize the black turning on the light.  

With the new fixture, both the fan and light functionality are controlled by the remote. Because of this, the fixture only requires a single ungrounded (hot) conductor to operate.  These types of fixtures are good for situations where a switch is not present/desired, but may be awkward when a switch is present.  As you've noticed, the fixture only operates when the switch is in the on position. The other switch in this situation would do nothing, since nothing is connected to the wire that it energizes.  

With this fixture you only require a single ungrounded (hot) conductor, because the remote routes power to either the fan, the light, or both depending on the setting.  The additional wire can be capped using a twist-on wire connector, or other approved method.

If you'd like to get the functionality you previously had, you'll likely have to bypass the remote on the fixture.  However, this is an atypical installation, and requires special knowledge.  This type of installation should only be performed by an experienced DIYer. 

The original wiring looked something like this...



The wiring with the remote unit looks something like this...



As you can see in the new wiring, the remote unit controls the current to the fan and light. Because of this, the second switched ungrounded (hot) conductor is not required.