Hayes put the Highlanders in front after a 10th-minute solo run, but his opener was cancelled out three minutes later by an excellent finish from Kelly for the hosts.
Kilmarnock looked the likelier side to clinch victory in the second half, with good opportunities falling to Rui Miguel and William Gros.
The points were shared but the draw, along with Motherwell's similar result at Hearts, ended Inverness' hopes of a top-six place.
Both sides made three changes to their starting line-ups. Home caretaker manager Kenny Shiels, taking charge of his first game at Rugby Park, gave starts to Rui Miguel, Liam Kelly and new arrival Ryan O'Leary in place of the injured Frazer Wright and Mohamadou Sissoko and benched David Silva, who featured in the previous Saturday's goalless draw at St Johnstone.
Inverness, playing their first game in three weeks following last week's late postponement against Celtic, brought in Kenny Gillet, Nick Ross and Aaron Doran for Eric Odhiambo and benched duo Russell Duncan and Grant Munro.
The opening stages were open and flowing with Inverness captain Richie Foran first to test a goalkeeper, but his speculative cross from the left byline on seven minutes was easy for Anssi Jaakkola in the home goal.
Jonny Hayes gave Inverness an early lead on 10 minutes - he robbed possession from Kilmarnock in midfield to embark on a pacy solo run, and despite having team-mates well positioned, finished the move himself by slipping the ball low into the right corner.
However, their advantage lasted just three minutes before Liam Kelly equalised with an outstanding finish, receiving the ball on the edge of the area and lifting a delicate lob into the net beyond a helpless Ryan Esson.
William Gros thought he had put Kilmarnock in front on 16 minutes, but his lob was ruled offside.
Kilmarnock carved some useful chances around the half-hour, the first when Craig Bryson and Alexei Eremenko linked well, despite a foul on the latter, for Bryson to set up Gros but his chance was booted clear by the outrushing Inverness goalkeeper.
Some more patient build-up from Kilmarnock led to a Rui Miguel header at the left upright which was cleared for a corner; Kelly almost swept home at the near post from the set piece, and Miguel then had another opportunity as Inverness failed to clear their lines.
Lee Cox had a chance for the Highlanders when he raced to the edge of the area for a snap shot which fizzed narrowly wide of the left-hand post.
Home goalkeeper Jaakkola was then inches away from gifting a bizarre goal to Inverness 10 minutes from the break - stooping to scoop up a harmless Aaron Doran free-kick under no pressure on the line, he somehow let it squirm through his grasp and had to scramble to claw the ball to safety.
Inverness brought on Gavin Morrison for Adam Rooney at the start of the second half, with Nick Ross moving into a forward role.
However, it was the hosts who had the first chance six minutes after the restart, Miguel powerfully heading a Garry Hay cross goalwards, but Esson made a comfortable save.
Miguel then successfully latched on to a long ball up the right and weaved his way into the box, cutting inside visiting defender Ross Tokely - but then wastefully ballooned his effort high over the bar.
The home fans had their heads in their hands when Gros passed up a glorious chance to put their side in the lead for the first time just before the hour with what would have been a well-worked goal.
Some precise passing play involving Hay on the left and Bryson in the centre allowed Alexei Eremenko to cross from the left byline for Gros, who somehow contrived to miss an easy close-range shot at the far post.
Newly-introduced substitute James Dayton tried his best to generate a Kilmarnock winner as the game entered its final quarter by going on a run from midfield; his shot was only parried by Esson to the right side of the box but the loose ball was booted to safety.
Bryson and Hay linked well once again for the latter to send in a deep cross from the left which the Inverness goalkeeper had to tip over for a corner at the far upright.
Kilmarnock looked the likelier candidates to find a winning goal in the closing stages, their best chance falling to David Silva straight after he was introduced two minutes from time, only for Esson to parry his shot.