How many watts will this amplifier need to dissipate?

A theoretically ideal Class D amplifier is 100% efficient and doesn't dissipate any heat. Page 15 of the datasheet shows that it's more than 80% efficient across most of the output range, so at 20W output, you'd be dissipating about 1W.

Normal music & voice has a peak-to-average power ratio of around 10, so with 20W (undistorted) peak output, the average output would be about 2W. (But of course, good audio amplifiers are designed to run at full power continuously.

So...you have to assume full power dissipation at all times.

The datasheet does NOT show that. The efficiency is 80% or more above about 1W. Below 1W, the efficiency falls-off.