Can I use a Class D amplifier as a summing amplifier?
Yes you can but not like that !

You treat the Class D amplifier as if it simply is a linear opamp. It is not.

Look in the datasheet of the TPA3116 and you will see that the application diagram does not show feedback configuration like in your schematic. Do you understand what a class-D amplifier actually is ? 

I'm also ignoring the fact that the TPA3116 has a balanced output for the loudspeaker, it is not (and cannot be) grounded on one side.

These "Tripath" amplifiers also have a sophisticated Analog to PWM thingy going on inside there, also ignoring that.

The fact that TI just bought Tripath instead of designing their own also means these amps are complex beasts.

But you're making this way too complex you can just combine the two audio input signals at the input of the power amplifier, feed each signal to the power amplifier's input via a 1 kohm resistor. There is no need to make a proper virtual ground summing point. The combined audio signal will simply be an average of the two and whenever I used that, that simply works.

So forget about feedback and especially feedback in combination with class D amplifiers.