Defeats to Sampdoria and Genoa had boosted the Nerazzurri's hopes of moving into the top six but they failed to take full advantage
Despite their inconsistent campaign, Inter can still seal a Europa League place in the closing weeks of the season, but failed to capitalise on Samp's defeat to champions Juventus on Saturday.
Roberto Mancini's men impressed in the opening 45 minutes as they took the game to their opponents, Mauro Icardi seeing the best chance thud against the post.
However, Inter's failure to break the deadlock nearly cost them after the restart as Chievo had more of an influence on proceedings, Samir Handanovic getting down well to deny Alberto Paloschi.
It was Handanovic who was the busier of the two goalkeepers in the closing minutes, much to the frustration of the home fans, as Inter were wayward with the chances that came their way.
Despite dropping two points, Inter climb one place to eighth and sit just two points off Sampdoria in sixth with four games remaining.
After going five matches unbeaten in the league the confidence within Mancini's side was clear to see as they made an encouraging start, creating two chances in the opening 10 minutes.
The hosts could have opened the scoring after two minutes as Icardi and Fredy Guarin combined well before the former found Rodrigo Palacio, but the offside flag denied him a seventh goal of the season.
A poor Chievo corner was nearly punished in a quick counter-attack five minutes later, Palacio dispossessing Ezequiel Schelotto and playing in Guarin, only for the midfielder to curl his effort wide.
Wave after wave of Inter attacks followed and Icardi thought he had netted 20 minutes in as he rose above two defenders to meet Juan Jesus' cross. His header beat Albano Bizzarri, but was kept out by the upright.
As Chievo struggled to get out of their own half, Inter were looking the more likely and, had Mateo Brozovic connected with a volley from the edge of the area, they would have gone into the break ahead.
The visitors posed more of a threat early in the second half as Riccardo Meggiorini and Paloschi both went close, before the latter stung the palms of Handanovic just past the hour.
Mancini looked to substitutes Lukas Podoloski and Xherdan Shaqiri to breathe life back into the game, but Chievo were growing in confidence while impatience began to creep into the hosts' play.
As whistles started to come from the stands, Inter desperately tried to create a moment of magic and, although chances did come for Shaqiri, Podolski and Icardi, they were unable to end the stalemate.