Rees could also sit out the following week's match against Scotland at the Millennium Stadium, with the Welsh Rugby Union reporting that his rehabilitation will take 'approximately two weeks'.
His absence from the trip to Dublin means Huw Bennett is likely to wear the No2 shirt at the Aviva Stadium after the Osprey deputised for Rees in the recent World Cup.
Rees was appointed captain of his country for the game against Australia in the autumn of 2010 and went on to lead Wales in the 2011 6 Nations.
A neck injury prevented him from travelling to New Zealand for last September and October's global gathering but the 31-year-old did return for the defeat to the Wallabies in Cardiff two months ago.
His late withdrawal is another big blow for Warren Gatland and co who have fitness concerns over another trio of key men.
Fly-half Rhys Priestland, blindside flanker Dan Lydiate and Lions centre Jamie Roberts are all doubtful for the opening weekend of 6 Nations action.
Gatland has delayed naming his matchday 22 until Friday in order to give his star men the best chance to recover.
Wales are already without second row duo Luke Charteris and Alun Wyn Jones and influential loosehead Gethin Jenkins.