Fernando Santos's men had scored 16 goals during their five-game winning streak across all competitions but could only muster two shots on target during a frustrating outing in Dublin.
Cristiano Ronaldo went the closest, though even the all-time leading scorer in men's international football could not break the deadlock.
The visitor, which had Pepe dismissed late on for two bookable offences, and Serbia both sit on 17 points with one game remaining as the latter travels to Lisbon for a winner-takes-all decider.
Andre Silva forced the first save of the contest from Gavin Bazunu before Seamus Coleman's excellent last-ditch block denied Ronaldo as the visitors dominated the early proceedings.
However, Republic of Ireland responded well and should have taken the lead on the stroke of half-time but Chiedozie Ogbene could only head over from close range.
Stephen Kenny's men continued industriously after the interval and almost punished Matheus Nunes' concentration lapse when Josh Cullen curled into Rui Patricio's hands.
Ronaldo, who later fired narrowly off target, then nodded wide following Silva's cross before the hosts saw Enda Stevens whip over from Ogbene's inviting delivery.
The visitor, which had Pepe dismissed for a second bookable offence after catching Callum Robinson with his hand, managed to hold on for a point despite the hosts thinking they had found a late winner through Matt Doherty, Will Keane's foul on Patricio cutting short the celebrations.