Fine Gael is to table a motion of no confidence in the Government in the Dáil next Tuesday.
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Fine Gael leader Enda Kenny said the election results represented a verdict of the people and he urged every deputy in the Dáil to consider their position in voting on the motion next week.
Mr Kenny claimed the results showed that the days of Fianna Fáil dominance were over.
Election results bore out earlier tallies - it was a bad day for Fianna Fáil and the Greens, with Fine Gael, Labour and Independents making significant gains.
This trend was also clear in the Dublin bye-elections, with Fine Gael's George Lee taking Dublin South and Independent Maureen O'Sullivan joining him in the Dáil with her victory in Dublin Central.
As the results become clear, the Taoiseach said he accepted that the people had spoken but insisted it was now time to move forward.
Brian Cowen said he knew some of the Government's decisions were unpopular but said they were necessary as part of a plan to fix the recession.
While Fianna Fáil's vote was slightly better than recent opinion polls, it was still an historic low, and the party dropped seats in councils all over the country.
The Greens also had a dismal day losing all its city and county council seats in Dublin, where most of its TDs are based.
Green Party TD Ciarán Cuffe told RTÉ News that his party's performance raised the question of whether it should be in Government.
But party leader John Gormley said the results are a clear indication that some Government decisions have been deeply unpopular.
Mr Gormley said his party must now reflect and listen to the anxiety of both its members and the public generally.
However, he said the question of possibly withdrawing from Government does not arise.
Fine Gael will be by far the biggest party in local government, while Labour is on course for its best ever result.
Labour Party leader Eamon Gilmore said it 'was a good day' for his party, with 'impressive performances' from Labour candidates in both Dublin bye-elections.
The Socialist Party and the People Before Profit Alliance have also performed well in Dublin, but Sinn Féin will lose seats in the capital.
