UPDATE: I'm now getting sound during Sonar MIDI playback but it seems to be double-triggering each note the Fantom plays. Is there a setting in Sonar or the Fantom that will remedy this?

Look at your local echo settings?

I've turned off Local Control on the Fantom X6 but I’m finding that with local control on the X6 turned off, I get no sound during Sonar MIDI playback.
 
Unless I turn MIDI echo off in Sonar, every time I press a key on the X6 or attempt Sonar MIDI playback, each MIDI note is triggered multiple times, resulting in somewhat of a MIDI note loop which can sometimes cause the X6 to hang/freeze. The only way I’ve found to stop this is turn off ‘Always echo the current MIDI track’ in Sonar but I’m not sure that’s really solving the problem?

only way I’ve found to stop this is turn off ‘Always echo the current MIDI track’ in Sonar but I’m not sure that’s really solving the problem?
 
No that is correct
 
What your doing is more complicated that a set up where your controller and your sound module are separate devices. 
So the loop back is going to happen. The easiest way to break that chain is turn off input echo. Or unplug the midi cable.
 

Do you set the Phantom in patch mode? or performance mode? FYI, To use multi channel / multi timbral you need to be in performance mode.