Why was Hastings imprisoned before the start of Richard III?
The issue at the opening of the play is who has sent Hastings to prison, rather than what he was imprisoned for. This is an important distinction, as it emphasizes the political nature of being jailed as it happens to the nobility in this play. Any charge can be trumped up against anyone; the infraction is not the point. The point is -- Do the powers that be want you out of the picture or in the picture?
At the opening of the play, Richard convinces Clarence that Hastings was sent to prison by the king's wife, Queen Elizabeth. He says:
Tis not the king that sends you to the Tower.
My Lady Grey his wife, Clarence, 'tis she
That tempers him to this extremity.
He calls the queen Lady Grey, since she was the widow of Lord Grey before she married the king. So, Richard has already begun to pit characters against each other in his bid for power, and one of his first moves in this direction is convincing Clarence that Hastings was jailed at the insistence of Queen Elizabeth.