The 30-year-old has made 24 appearances for the Tigers since joining the club from Toulouse last summer and has become a favourite with the Welford Road supporters.
Director of Rugby Dean Richards explained that Tournaire's departure was not rugby-related.
"He has domestic problems in France, which I don't want to go into," said Richards. "I have sat down with Franck and his agent and we agreed that it's in Franck's and his family's best interests that he returns to France.
"I feel sorry for Franck and he's very upset about the situation himself, but at the same time it's vitally important that he does go home, bearing in mind the situation.
"I'm disappointed he's not staying. I have been pleased with Franck – he's got some great footballing skills and he's demolished some of the packs that he's played against. It was going very well. It's just very disappointing from his point of view and from ours that he's got to go back.
"Everybody at the club will be sad to see him go. He has fitted in extremely well here and made some good friends. I will certainly keep in touch with him and we all wish him well in the future."