The world number five, who made the last four at SW19 last year, bowed out of the Wimbledon warm-up competition at Queen's with a 6-7 (3/7) 6-3 6-7 (5/7) loss to world number 69 Dodig.
His disappointment was furthered by the news that he may have broken the little finger on his right hand after slipping in the sixth game of the third set.
"I think it's serious," said Tsonga, who had his hand placed in an ice-bag during his post-match press conference.
"I will have a scan, but it sounds very bad.
"I felt like I broke something or I strained it when I fell.
"It's only a finger but it's tough because I have to play with my hand.
"At the moment I don't know what is it, but I will stay positive and I will see."