The 20-year-old right-back went to hospital with Gulls physiotherapist Gareth Law on Wednesday and will not be involved on the pitch until both clubs are sure that he has made a full recovery.
"When it's chest or heart-related things you've just got to make sure," Gulls manager Kevin Nicholson told BBC Sport. "He's going to go up to St George's Park to have a thorough set of tests, just to make sure there's nothing.
"I'm pretty sure that it's nothing major, but until ourselves and Bournemouth are comfortable that he's 100% then, unfortunately, he's not available to us.
"He just woke up in some discomfort in his chest. It's probably a virus, and these things can show up like that, but we've just got to make sure he's right and so we wish him a speedy recovery and look forward to getting him back."
Lee, who has made two FA Cup appearances for Bournemouth, is in his second loan spell with the National League club and has made 11 appearances for them in total.