The Doctor Who episode “Time Heist” is a lot like Ocean’s 11 mixed with the myth of the Minotaur, with the twist of saving the Minotaur.
In the United States, a person commits “bank robbery” if they enter a bank with the intent to commit any felony, or any larceny, or take any property from a bank belonging to another person that is at least $1,000 in value. 18 USCS § 2113(a) and (b).
The Doctor’s ultimate reason for breaking into the bank is to rescue the Teller and his mate who had been imprisoned in Director Karabraxos’ private vault. It is not the fact the Doctor saved the Teller and his mate that constituted bank robbery, but taking the items to restore Psi’s memories and medicine for Saibra’s mutation. Unless those items belonged to the elderly Ms. Karabraxos who effectively hired the Doctor to conduct the heist, those two items were under the custody and control of the bank, most likely over a $1,000 in value, and would meet the plain language of the statute.
However, if Ms. Karabraxos owned those items, then she could have properly paid the Psi and Saibra for saving the Teller from her own bank. Moreover, the Doctor and his companions could not be breaking into a bank if they had permission from the owner, even if she was in the future.
Additionally, as the Bank of Karabraxos was holding one creature in slavery, plus one other imprisoned as leverage, this was one bank that was operating outside the bounds of the law. Moreover, erasing the minds of suspected criminals and putting them on display showed the bank was operating under its own legal system. There is no way such an organization could be insured by the FDIC or be recognized as a lawful bank.