Help Extract Animals from Red Tape Act of 2021 or the HEART Act of 2021 This bill modifies notice requirements in a civil forfeiture proceeding that involves the seizure of an animal involved in animal fighting. Specifically, the government must notify interested parties of an animal's seizure within 30 days (currently, 60 days). The notice period may be extended only if a delay is warranted after considering certain factors, including the cost to the government of providing shelter for the animal, the health of the animal and the effect of a delay on its rehabilitation, any increased risk that a delay could necessitate the euthanasia of the animal, and any psychological harm incurred by the animal's owner due to the delay in disposition. Additionally, the bill authorizes amounts in the Department of Justice Assets Forfeiture Fund to be used to pay for costs associated with a seized animal, including transportation, shelter, care, veterinary services, and humane euthanasia.