Victory over Brazil's Tiago Camilo would have put the 29-year-old Scot into a play-off for one of the two bronze medals available.
However, Camilo, the Brazilian 2007 world champion, held on to his lead after going ahead in the second minute of the contest with a throw for a wasa-ari.
"This guy is world champion and a great judo player," said Burton. "It was a close one but not the most exciting judo I've ever played that's for sure. It's the Olympic Games and every match is a test.
Video - Burton bows out
"I had a slow start - first matches are always nervy - it was scrappy all day but it doesn't have to look good to win.
"Mostly, to get an Olympic medal you have to come up with the big throws - that's not what I did today - but it wasn't for a lack of trying."
Camilo went on to win bronze after beating Guillhaume Elmont of the Netherlands.
Ukraine's Roman Gontyuk added Beijing bronze to the silver he won in Athens four years ago as he beat Mongolia's Damdinsuren Nyamkhuu.
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