What do hummingbirds do at night?

In a word, they fast. 

But don't they have to eat continuously to prevent starvation? And if the night is more than a couple of hours long wouldn't they starve? 

Yes, that's true, but they have a clever trick they can use to make their energy supplies last the entire night. They allow their metabolism to slow and their body temperature to drop, a condition called torpor. It's like hibernation, but it only lasts overnight instead of over several months. When daylight approaches they spontaneously warm up by shivering their flight muscles. When they reach their normal operating temperature they take off in search of food.

Twenty-five years ago I was fortunate to spend a couple of days banding hummingbirds under the supervision of the late Bill Calder. Bill had set up mist nets (nets with a mesh so fine that it is not noticeable to birds) in a flower-filled, montane meadow at ~9000 ft. elevation in the Elk mountains of western Colorado. We arrived to set up the nets early in morning, just as the sky was lightening but before the sun was up. As we walked through the meadow we could hear the buzzing flight sounds of numerous hummingbirds, already actively foraging for nectar in the very chilly early morning mountain air. 

As soon as we deployed the nets we started to hear the distress calls of hummingbirds trapped in the mesh. With our head lamps on we rushed to find the captured hummers and gently extricated the birds from the mesh. We then placed them in bags made from Bill's wife's nylon hosiery, and took them to Bill who weighed them, checked them for bands and, if they were unbanded, clamped a tiny, numbered band around one leg. They were then released. From capture to release took at most five to ten minutes. We worked continuously for several hours until the rate of capture fell to zero. By an hour or two after sunrise the ravenous hummers had drunk their fill and could slip into their daily routine patrolling their trap lines and stocking up on enough nectar to last them through the next night.