How many watts will this amplifier need to dissipate?

fungus:

MarkT:
Also remember most audio signals are nowhere near maximum amplitude most of
the time, its only the percussive peaks that get up there, so in practice the dissipation
will be a lot less than the maximum.

Not in a class D amplifier.

While I'm inclined to agree with you because what you said earlier seems to make sense, if that's true, then how do you explain the efficiency vs output power graph in the top center of page 8 in the datasheet:

That graph is for a 12V input into a 4 ohm load. The line at the bottom shows the power dissipated vs the output power. The power dissipated clearly increases in a linear manner with the output power, which varies with an increasing duty cycle.

It's not even a little change either, it's a big change in power dissipation. And that directly contradicts your suggestion that the dissipation remains constant.