No, a hummingbird’s heart will not stop if it stops moving. You’re possibly thinking of Spoink, which is a Pokemon that supposedly powers its heart by bouncing, and as such can’t stop moving. Hummingbirds don’t sit often because they’re busy looking for food, but they can and do sit. The females sit on eggs in nests, after all, and they do have to sleep. 

Unrelated fun fact: the primary Aztec god of war would take on the form of a hummingbird, and the souls of the bravest warriors were said to turn into hummingbirds in order to join him after death, presumably because every hummingbird is approximately four grams of pure concentrated asskicking which fears no man nor beast and will gladly throw down with somebody seventeen thousand times their size if offended.

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