According to this source (emphasis and links mine), Tucci's portrayal was inspired by at least three other presenters:
Tucci plays flamboyant presenter Caesar Flickerman in the hit film franchise and his portrayal was inspired by American entertainer Wayne Newton, fictional presenter Jiminy Glick, and camp Irish chat show host [Graham] Norton.
In an appearance on Norton's programme this week (ends12Apr15), Tucci admitted the host had been a big influence on the development of Flickerman, telling him, "When you create a character, you steal and cull from all sources. He's a cross between Wayne Newton and Jiminy Glick and there's a little piece of you Graham - the intelligence!"
And according to another interview, Tucci's inspiration for the look of Caesar (as opposed to the performance, presumably?) was based on George Hamilton and Karl Lagerfeld.
The only reference I can find to any inspiration by Suzanne Collins in writing her character was in the notoriously unreliable Hunger Games Wikia
, which claims "one might think his last name to be a reference to David Letterman. However, his portrayal is perhaps closer to Leno, which if intentional would mean that Ms. Collins is subtly mocking Jay." However, this may well be completely made up as an excuse to get a nice bit of wordplay in ("mocking Jay"). I definitely wouldn't put any faith in this totally unsourced claim.