Mr Zelenskyy wanted to make an unexpected video appearance during the final in Liverpool to implore the event's global audience of millions to continue backing his country in its fight to repel its Russian invaders.
But the European Broadcasting Union, an alliance of 112 member organisations that organises the annual contest along with the host broadcaster, which this year is the BBC, has refused.
The EBU said Mr Zelenskyy had "laudable intentions" but that "regrettably" his request was against the rules.
"The Eurovision Song Contest is an international entertainment show and governed by strict rules and principles which have been established since its creation," an EBU representative said.
"As part of these, one of the cornerstones of the contest is the non-political nature of the event.
"This principle prohibits the possibility of making political or similar statements as part of the contest.
"The request by Mr Zelenskyy to address the audience at the Eurovision Song Contest, whilst made with laudable intentions, regrettably cannot be granted as it would be against the rules of the event."
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The BBC has said the broadcast of this year's contest in Liverpool is expected to be watched by more than 160 million viewers worldwide.
Ukraine's Kalush Orchestra won last year's contest but, owing to the Russian invasion of the country, hosting duties were instead awarded to the runner-up, the UK.
"No fewer than 11 Ukrainian artists, including last year's winners Kalush Orchestra, will be performing," the representative said.
"Additionally, 37 locations around Ukraine will feature in the short film postcards that introduce each of the participating artists before they take to the stage.
"We believe that this is the best way to reflect and celebrate Ukraine's Eurovision Song Contest win and show we are united by music during these hard times."