Do beignets have to sit overnight?
Warning: this answer isn't about traditional beignets, but about New Orleans beignets, which are actually made from yeast dough, like donuts. It refers to this recipe which the OP is most likely using. 



For yeast dough, the difference between long and short rising times is that longer times give you more flavor development. Beignets are so chock-full of sugar and fat that they don't need the subtle flavoring of long fermentation. They can be made with very short rising times. 

I don't know why the recipe tells you to refrigerate overnight and then forgets to tell you that the dough should come back to temperature. There are some rich doughs which are baked from cold in order to hold their shape (brioche and other butter-rich relatives), but due to the physics of deep frying, you don't want fridge-cold dough in your pan. But there is surely no need to get it thoroughly cooled; the recipe is probably advising the long stay for fermentation or convenience reasons, not because you need chilled dough. One possibility which comes to mind is that the soft dough might be a bit hard to roll when warm, but a combination of non-stick mat, clingfilm and flouring usually takes care of that. 

If you keep your dough out of the fridge, it will be ready much earlier, and you can fry it as soon as it is properly risen. But because this recipe has such an insanely high yeast ratio (equivalent to 7.5% fresh yeast in baker percentages), you have to watch it closely, it might rise very soon.