This seems like something that should be true, but I think I remember seeing a Mythbusters episode where they decided it didn't make a difference. That show was more about entertainment than science, so I wondered if there was a more rigorous study done? I've definitely seen splashes of water(?) come out from flushes so that alone seems to argue for closing lids.
I think someone urgently needs to come up with one of these solutions:
The foot-operated lid;
The toilet with flush and suction;
The Jedi throne (a Jedi-style toilet lid activated by hand movements)
and lastly
The Terminator (a time-activated flames of hell) solution. The time-activated mechanism locks the toilet door after the user leaves and burns the entire compartment at solar flare temperatures.