There's no evidence I've found that the line comes from Warner himself, from wikipedia:
Plummer said that while he greatly enjoyed the part of Chang, he regretted that David Warner (Chancellor Gorkon) got what Plummer considered to be the best line in the film, "You've not experienced Shakespeare until you've read him in the original Klingon". Academics have suggested several interpretations of this line, some seeing it as a joke, others as something more serious.
— "How Christopher Plummer Became One of the Best Villains in Star Trek Movie History"
— Kazimierczak, Karolina (2010). "Adapting Shakespeare for "Star Trek" and "Star Trek" for Shakespeare: "The Klingon Hamlet" and the Spaces of Translation"
In fact, from the The Klingon Hamlet wikipedia page there's this very convincing quote
The film's director Nicholas Meyer said the idea for having the Klingons claim Shakespeare as their own was based on Nazi Germany's attempt to claim William Shakespeare as German before World War II. A similar scene appears in the wartime British film "Pimpernel" Smith (1941) in which a German general quotes Shakespeare, saying “'To be or not to be', as our great German poet said." The idea had also already been used by Vladimir Nabokov in his novel Pnin, the eponymous hero of which taught his American college class that Shakespeare was much more moving "in the original Russian."
Emphasis mine.
So it looks less like the actor ad-libbed the line, and more that Meyer was playing off of real-world examples of people claiming Shakespeare's work as coming from their own country.
The film the line comes from, was already inspired by real-world politics, as much of early Trek was:
Nimoy visited Meyer's house and suggested, "[What if] the wall comes down in outer space? You know, the Klingons have always been our stand-ins for the Russians ...".
— Nicholas Meyer: Gorkon is Gorbachev".
So this seems more plausible an inspiration for the line.