Broadchurch star Tennant plays the debauched lead character in Don Juan in Soho.
Patrick Marber's play - first seen in 2006 - has been updated and now includes references to US President Donald Trump and vlogging.
Speaking after Tuesday's opening night, Tennant revealed that Marber had written in a line about the Daily Mail's controversial "Legs-it" headline, but it didn't make the show.
"We chickened out at the last second!" he told the BBC.
"We are sprinkling occasional contemporary references as we go. We add them on the day. Prince George is in his third day now. George Osborne lasted for a weekend. Donald Trump is in there."
Tennant added: "Obviously it has to relate to Don Juan and his world view. He's got this fury about the hypocrisy of the world that he's in.
"The world keeps presenting itself with fresh material and we can't really resist."
Marber's play is a loose adaptation of Moliere's 17th Century tragi-comedy Don Juan that transports the legendary womaniser to contemporary London.
Tennant's character - known as DJ - describes himself in the play as the "Dalai Lama of debauchery".
At one point his servant Stan, played by Adrian Scarborough, urges the audience: "Please don't be charmed, he's not a lovable rogue..."
"It's a delicious role," Tennant said, adding that he hadn't done a play that wasn't Shakespeare in about 14 years.
"I adore Shakespeare - I feel like it's in my DNA - but I'm just loving doing something that is this contemporary and that has this bite and extraordinary language."
Don Juan in Soho is at Wyndham's Theatre in London until 10 June.
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David Tennant says his latest play is being constantly tweaked to feature topical jokes - because the news keeps producing such good material.