Newly-crowned US Open champion Rose will instead partner defending champion Ernie Els and American Brandt Snedeker at Muirfield, the trio teeing off at 9:11am on Thursday.
Rose will be followed onto the course by close friend and Ryder Cup team-mate Ian Poulter, who has American duo Keegan Bradley and Billy Horschel for company at 9:22am.
And world number two Rory McIlroy is due out at 9:44am with Scottish Open winner Phil Mickelson and Japan's Hideki Matsuyama.
Woods has to wait until 2:45pm to begin his challenge for a 15th major title, the world number one drawn alongside 2010 champion Louis Oosthuizen and Northern Ireland's Graeme McDowell, winner of three of his last eight events.
The first shot of the 142nd Open Championship will be struck at 6:32am by Australian veteran Peter Senior, with Scotland's Lloyd Saltman and England's Oliver Fisher making up the first group. The last group does not get under way until 4:13pm.
Rose finished 13th in the Travelers Championship immediately after his US Open win, but withdrew from the AT&T National the following week due to fatigue.
The 32-year-old said he feels refreshed after three weeks off and is "itching" to return to competition, especially with the chance to join Bobby Jones (twice), Gene Sarazen, Ben Hogan, Lee Trevino, Tom Watson and Woods as the only men to win the US Open and Open Championship in the same year.
Woods has not played since finishing 32nd in the US Open at Merion, where he exacerbated an elbow injury first suffered during his victory in the Players Championship at Sawgrass in May.
The 37-year-old wrote on his website at the weekend that he was still taking anti-inflammatory medication but was "very confident" that the injury to his left elbow would not hamper his chances of a first major title since the 2008 US Open.
Woods was on course for the Grand Slam the last time Muirfield staged the Open in 2002 after winning the Masters and US Open, but saw his hopes blown away by a third round of 81 in terrible weather.
McIlroy missed the cut in his only event since the US Open, admitting he had yet to find the right driver and putter since his controversial multi-million pound switch to Nike in January.
But after a practice round on Sunday, the 24-year-old told reporters he had found the right driver at least.
"I went over to London after the Irish Open, tried a few drivers with different settings and as soon I hit it I said, 'This is the one.' It feels great," he said.
The big-hitting trio of Bubba Watson, Nicolas Colsaerts and Dustin Johnson, who were also drawn together at Merion, tee off at 8:44am and are followed by Nick Faldo, Tom Watson and Fred Couples, who can boast eight Open titles between them. Two of Faldo's came at Muirfield in 1987 and 1992.
Masters champion Adam Scott, who blew a four-shot lead with four holes to play at Lytham last year, is alongside Matt Kuchar and Luke Donald at 2:12pm, with Lee Westwood, Charl Schwartzel and Sergio Garcia in the preceding group.