He insisted that he had never stood to profit from the booking and the Conservatives insisted no rule had been broken.
But Labour said the incident, later reported by the Independent, was a "sign of systemic moral failure" in the Conservative Party leadership.
It was advertised as a "one-of-a kind opportunity to advance your lifestyle" and also included two days of tennis, accommodation at the Dorchester hotel in Park Lane and a drinks reception with former tennis pro John McEnroe.
Lord Fink returned to his post as treasurer earlier this year following the resignation of Peter Cruddas, who was recorded telling undercover reporters that major Tory donors would get privileged access to David Cameron.
The Tories said Lord Fink was no longer sponsoring the dinner after it was brought to his attention that it may be in breach of the rules. A source said: "If there was a mistake then it was a completely innocent one."
Lord Fink said: "This event has been held in the Lords on several occasions in the past and I agreed to sponsor it this year in return for a sizeable donation to a major hospital.
"I have no commercial interest in Amex and there was never any question of me profiting from sponsoring it."
A Conservative Party spokesman said: "Lord Fink cancelled his sponsorship of the event at the beginning of last week as soon as he received advice that there might be an issue with it."
Jon Trickett MP, Labour's Shadow Minister for the Cabinet Office, said: "This is a flagrant attempt to breach the rules of Parliament. It leaves a very nasty taste to suggest that you can buy access to the House of Lords, and it cheapens democracy.
"Lord Fink, who is a major donor to the Tories should consider his position as Treasurer of the Conservative Party and if he refuses to do so the Prime Minister should sack him.
"To have one Tory Treasurer in trouble for appearing to sell access to Number 10 is bad enough, some might think that now to have a second Treasurer apparently in breach of the code of conduct is a sign of systemic moral failure in the high command of the Conservative Party."
