The Irish needed to beat Denmark in Dublin on Monday night to leapfrog the Danes for a Euros spot but had to settle for a 1-1 draw.
In a defensive contest, the visitors went ahead in the 73rd minute through Martin Braithwaite.
Matt Doherty equalised in the 85th minute as Ireland pushed numbers forward but couldn’t find a winner.
Switzerland proved too good for Gibraltar 6-1 to seal top spot in Group D, one point ahead of Denmark with Ireland a further three points back.
Ireland failed to win in their final four group matches and will need to progress via the play-offs.
In Group J, Italy completed a perfect campaign with an emphatic 9-1 victory over Armenia.
Ciro Immobile and Nicolo Zaniolo both scored doubles in the win.
Finland, who had already sealed their qualification, lost 2-1 away to Greece despite Teemu Pukki opening the scoring.
Bosnia-Herzegovina rounded out a disappointing group stage with a 3-0 win at Liechtenstein.
Spain and Sweden finished Group F with comfortable 5-0 and 3-0 wins respectively over Romania and Faroe Islands. Norway beat Malta 2-1 in the other game.