Can the FFT reduce its computation load for specific sparse data patterns?
Can the FFT reduce its computation load for specific sparse data patterns?

Sort of but only of it's very sparse. You can always NOT use an FFT but implement the inverse FFT directly and omit the 0 terms from the sum.
We can guestimate the break even point. An FFT takes about 10 operations per butterfly and for a length 256 FFT you have 8 stages with 128 butterflies each, so you end up with about 10000 operations per FFT
For a direct inverse DFT you'll have for each real output sample about 3 operations per non-zero frequency coefficient. In this case the break even point comes out to be 13 or thereabouts.
Your case of 16 is probably close to a wash, so I wouldn't bother.