Believe it or not they could, at least for a very short time. But only under very specific circumstances. Here are some specific ways that fireworks could cause problems while they’re actually occurring:
Shaking your mast. If the antenna or dish isn’t tightly mounted on a mast, or if the mast isn’t really well anchored, then the percussion of the explosions could actually vibrate your antenna or dish out of alignment at least temporarily. But you’d have to be pretty close for this to happen.
EMI during explosion It’s possible but unlikely for a fireworkd explosion to cause a measurable amount of electromagnetic interference when the actual explosion happens. This would be a very very short interruption though, and not likely to be a real cause for concern.
Something in the clouds Of all the possible causes of interference, this one is actually the most likely. The bright colors from fireworks come from different metals that are mixed in with the gunpowder. When the gunpowder explodes, the metals oxidize instantly giving you bright flashes of color. However, if some of the metal does not oxidize it could remain suspended in a fine mist for a period of time and a cloud of metal dust could conceivably float between you and the broadcast source. It’s still not really likely but it’s the scenario that’s most likely of all of them.
Intereference from unauthorized use Now, no one seems to know how this happens but it seems like an awful lot of people come into possession of illegal or unauthorized fireworks around this time of year. In much of the west, all home fireworks are illegal except “safe-and-sane” types that really ought to be called “boring-and-boring.” Yet there’s quite a lot of the old-school firecrackers out there and they’re likely to cause problems with your reception if for example you throw them up in the air and they land on an antenna or dish. In fact rockets and firecrackers will cause a lot of damage if you use them wrong which is why they’re illegal in a lot of places.
But… as for the myth that fireworks will cause this massive jolt in your reception that you won’t be able to recover from, that’s not true. If there’s anything it’s likely to be very quick and not worth worrying about.