The two-and-a-half minute-long clip - aired in Australia - also features two Roman soldiers asking to pose for a selfie photograph with Christ on the cross.
Several complaints have been made to the Australian Ad Standards body, which is considering whether to stop the material from being broadcast.
The ad prompted dozens of comments on Facebook, including one from Aimee Redding who said: "I am not easily offended but I was deeply offended and saddened by this ad.
"I believe it should be removed and an apology issued."
While Lynette Shryane said: "One day every knee will bow and tongue confess that Jesus Christ is Lord. It won't be such a joke then. Very poor taste."
But fellow social media user Sebastian E'Silva disagreed, writing: "Excellent ad, actually. I love the line delivery, editing and way it's been filmed.
"The fact that it's going to tick off humourless fundies [a term referring to Christian fundamentalists] while gaining attention for a good cause is the cherry on top."
The film was released as part of Dying to Life - an initiative launched in response to concern Australia "lags behind similar countries in organ and tissue donor registration numbers".
Director, Richard Bullock said: "I wanted to deliberately provoke a conversation in homes around the subject.
"I thought it would be amusing and relevant to find that the nicest and kindest man who ever lived - Jesus, wasn't aware that his organ donor status was no longer on his license.
"Once I started writing I realised that the complexity of the Australian Organ donation could be explained.
"In the end Jesus donating his organs is exactly what I think Jesus WOULD do".
Meanwhile, Michael Stead, the Anglican Bishop of South Sydney, told the Ten Daily news website that some viewers "might be offended by the Monty Python-esque humour" but "it uses the self-sacrificial example of Jesus to communicate an important message."
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