The poll of 1,650 Britons by YouGov found 56% would quit the competition.
And it found those who backed Leave in the EU referendum were most likely to want to drop out, with 76% to 21% in favour of quitting. Remain voters were 65% to 35% in favour of competing.
The UK first entered Eurovision in the second year of the contest in 1957 and has won five times - the last in 1997.
That victory for Katrina and the Waves with Love, Shine a Light, followed wins by Sandie Shaw (Puppet on a String, 1967), Lulu (Boom Bang-a-Bang, 1969), Brotherhood of Man (Save Your Kisses for Me, 1976) and Bucks Fizz (Making Your Mind Up, 1981).
However, since 1997 the UK has finished last three times - in 2003, 2008 and 2010 - and has failed to make it into the top 10 for the past seven years.
This year's contest will take place in Kiev on Saturday, following Jamala's win for Ukraine last year singing 1944.
The UK will be represented by former X Factor contestant Lucie Jones, who will sing her ballad Never Give Up On You, but according to YouGov only a fifth of Britons (22%) are intending to watch the show.
Of those planning to watch, 26% of people said their reason for watching was the "amusing commentary", which will again be provided by Graham Norton, 19% are planning to make fun of the show and just 9% say they tune in because they like the music.
The UK would make its mind up to leave the Eurovision Song Contest if there were a referendum, a survey suggests.