"My tendon snapped once again in Montpellier," Tsonga said.
Tsonga said he will miss the rest of the season, hence has ruled himself out of the December 3-5 Davis Cup final in Belgrade.
Tsonga had also not taken part in France's quarter and semi-final wins over Spain and Argentina.
Tsonga, the 2008 Australian Open runner-up whose career has been blighted by injuries, was also sidelined for three months after Wimbledon with a knee problem.
"I am dejected because the Davis Cup is a kid's dream. It's the biggest setback in my career along with having to pull out of the Olympic Games," said Tsonga.
"The injury is less serious than last summer but the doctors told me to end my season.
"I could have taken anti-inflammatories but I don't know how I would have endured the pain. It would have been dishonest to my team mates."
Tsonga, however, is hoping Forget will still take to Belgrade so that he can support his team mates.