New coach Graham Rix has now drawn three off his first four games as Hearts' form slumps at a vital stage of the season.
But it could have been worse for Hearts because Inverness were the better side and created the clearer chances.
Barry Wilson had a strong second-half penalty appeal waved aside when the ball struck Robbie Neilson's hand.
The Highlands side also had a goal disallowed when Ross Tokely fired home but was given offside.
Hearts played Paul Hartley in a deeper role and lost forward momentum in midfield and the best chances of the first half fell to the visitors.
Ian Black volleyed wide and, after a great shot from Craig Dargo, player-boss Craig Brewster fired the rebound over the bar from only ten yards.
Hearts' best chance of the first half fell to Deividas Cesnauskis when he exchanged a one-two with Hartley but then mis-kicked in front of an open goal.
The home fans became more frustrated but again Inverness continued to dominate and Craig Gordon pulled off a fine save to keep out Black's 25-yarder.
The Scotland international keeper was also at his best to tip-over a spectacular long-range effort from Brewster.
But with a difficult trip to Ibrox to come next week Hearts' title challenge is in danger of fading.