Inverness heaped pressure on the rock-bottom visitors, but wayward finishing and some fine saves from Jamie MacDonald forced Caley Thistle to settle for a point.
Both sides threatened in the early stages as Ryan Stevenson's drive was turned away by Inverness keeper Dean Brill, while Caley went close at the other end when Marley Watkins shot past the post.
Inverness began to get a grip on the game and David Raven's attempted cross hit the face of the bar before MacDonald kept out Greg Tansey's header.
Jordan McGhee helped out McDonald with a last-ditch clearance and Caley endured more frustration as Watkins and Billy McKay both struck the woodwork within a few minutes.
Inverness would strike the frame one more time in the first-period as Tansey thumped the ball against the post to prompt even more groans from the home fans.
Hearts were living dangerously, but they almost snatched a goal after the break as Jamie Hamill sent two shots just off target before Stevenson stung Brill's palms.
But Inverness regained control and MacDonald again held them at bay, scrambling away Tansey's shot.
As the game entered the closing stages, the Jambos could have snatched an undeserved winner but Hamill blazed over the bar.