We were at Disney during what we now refer to as "monsoon week" in May 2009. It was pouring rain pretty much every night that week except for Friday. We were staying at Pop Century that year and could see the fireworks from Illuminations & Fantasmic going off every night. There was no way I was standing in a torrential downpour to watch fireworks so we only went to Illuminations on the one night it wasn't raining. As far as I know, Wishes also happened every night that week.
In June 2011, it was raining so hard (no lightning) that after 2 delays, they did end up cancelling the Main Street Electrical Parade that night and Tinkerbell didn't fly, but Wishes did go off on time (in the rain). And last summer we were at DHS on a rainy/stormy night -- the Frozen fireworks and Fantasmic were both delayed until the rain stopped. We got about 10 minutes into Fantasmic when it started pouring again - no thunder or lightning, but heavy rain. We decided we didn't need to sit there in that mess (we have seen Fantasmic several times) so we got up and left. By the time we got down the ramp out of the theater, they announced that they were cancelling the rest of Fantasmic for the safety of the performers. We kept walking very briskly and made it to the bus stop ahead of the crowd. In fact, there were only 12 people on our bus. Our driver basically closed the doors and pulled away from the stop as soon as we were on the bus. While there was no one else at the stop at that time, I'm sure there were plenty of people in the huge mob pouring out into the bus loading area who would have gotten on our bus.
We've done the Illuminations Fireworks cruise a few times and our captain has always said that while Illuminations will still happen if it's just raining, if there is lightning close by it will get delayed and even cancelled (not just the cruise, the actual fireworks). They said it doesn't happen often, but it does happen.