The Scotland international was forced out of Friday's hard-fought league win over Benetton Treviso after just 22 minutes, with an x-ray revealing he had fractured his shin.
A MRI scan has since shown further damage to his ankle ligaments and Visser will now require surgery as a result.
Visser will miss the remainder of Edinburgh's pool games in December and January.
A try scorer in the opening round win over former champions Munster at Murrayfield and the following week's defeat at Perpignan earlier this month, Visser won't be available for next month's double header with Gloucester or the Round 5 and 6 ties at home to the Catalans giants and away to the Irish heavyweights at Thomond Park.
He should be fit to return should Edinburgh make the knockout stages for the second time in three seasons, though, with the quarter-finals of the Heineken Cup taking place on the weekend of April 4-6 2014.
"This is really disappointing news for Tim," said Edinburgh head coach Alan Solomons.
"He's an asset for Edinburgh Rugby and Scotland. We all wish him a speedy recovery and look forward to seeing him back scoring tries again soon."
Visser will also sit out all three of Scotland's autumn Tests against Japan, South Africa and Australia, as well as up to 10 matches in the RaboDirect PRO12.
He faces a big battle to play more than just a bit part in the RBS 6 Nations, with Scotland's games against Ireland, England and Italy all taking place before the four-month mark of his recovery process.