Scott Fox made saves to deny Hearts' Tony Watt and Conor Sammon before the interval.
The hosts continued to dominate after the break but their ability to hit the target waned.
County, who threatened through Andrew Davies, finished with 10 men after Ian McShane received a second booking for a tackle on Faycal Rherras.
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Fox was the busier keeper and he was on top form as he produced a sequence of top-class saves, particularly in the first half.
He reacted well to turn striker Watt's shot from 16 yards over and held headed efforts from Callum Paterson and Igor Rossi.
Sammon was denied twice by the agile stopper as Hearts continued to press.
At the other end, a Martin Woods free-kick was flicked, in an attempt to clear, by Paterson against the head of Davies and the ball had to be scrambled clear by the Hearts defence.
Some of the visiting players claimed the ball had crossed the line but referee Alan Muir was unconvinced.
Alim Ozturk was relieved of the Hearts captaincy duties earlier in the week with the armband passed to midfielder Perry Kitchen.
Having been kept out of the side because of the form of John Souttar, Ozturk got his opportunity after half-an-hour when the former Dundee United defender had to limp off injured.
Hearts' dominance in possession continued into the second period but they found it harder to test Fox with Sam Nicholson clipping the outside of the post and Rherras firing over.
The Staggies, who have never won at Tynecastle, lost a man when McShane - earlier booked for a foul on Nicholson - caught Rherras late and was promptly dismissed.
But the Highlanders held out to earn a point and their third draw in a row.
Hearts head coach Robbie Neilson: "A disappointing result but the first 60, 70 minutes the performance was great. Again we created loads of chances but we are not taking those chances as we should be.
"It's that final little bit, if we had done it today we would have another three points on the board.
"It was poor finishing from our point of view. The performance is the key for us. If the performance is there then the three points are there. But today similar to last week the three points weren't there."
Ross County manager Jim McIntyre: "If you take a point away from here it's never a bad result. That goes for any away ground in the Premiership, it's a tough league.
"We had Scott Fox to thank for a couple of good saves in that first period.
"We can't be disappointed with what the boys gave us today - I thought they were excellent and showed a real togetherness once we went down to 10 men.
"The Andrew Davies effort had not crossed the line. I've seen it back and as much as I would have wanted it to have gone over the line, it has not."
Match ends, Heart of Midlothian 0, Ross County 0.
Second Half ends, Heart of Midlothian 0, Ross County 0.
Attempt missed. Alim Ozturk (Heart of Midlothian) right footed shot from the right side of the box misses to the right.
Foul by Faycal Rherras (Heart of Midlothian).
Tony Dingwall (Ross County) wins a free kick on the left wing.
Attempt missed. Callum Paterson (Heart of Midlothian) left footed shot from the centre of the box is too high.
Marcus Fraser (Ross County) is shown the yellow card.
Corner,  Heart of Midlothian. Conceded by Marcus Fraser.
Tony Watt (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Paul Quinn (Ross County).
Arnaud Djoum (Heart of Midlothian) is shown the yellow card for a bad foul.
Foul by Arnaud Djoum (Heart of Midlothian).
Marcus Fraser (Ross County) wins a free kick on the left wing.
Arnaud Djoum (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Michael Gardyne (Ross County).
Second yellow card to Ian McShane (Ross County) for a bad foul.
Faycal Rherras (Heart of Midlothian) wins a free kick on the left wing.
Foul by Ian McShane (Ross County).
Attempt missed. Tony Watt (Heart of Midlothian) right footed shot from the right side of the six yard box is just a bit too high from a direct free kick.
Jamie Walker (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Michael Gardyne (Ross County).
Ian McShane (Ross County) is shown the yellow card for a bad foul.
Sam Nicholson (Heart of Midlothian) wins a free kick in the defensive half.
Foul by Ian McShane (Ross County).
Substitution, Ross County. Jay McEveley replaces Andrew Davies because of an injury.
Substitution, Heart of Midlothian. Bjorn Johnsen replaces Conor Sammon.
Igor Rossi (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Tony Dingwall (Ross County).
Jamie Walker (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Ian McShane (Ross County).
Corner,  Ross County. Conceded by Igor Rossi.
Foul by Arnaud Djoum (Heart of Midlothian).
Michael Gardyne (Ross County) wins a free kick in the defensive half.
Attempt missed. Perry Kitchen (Heart of Midlothian) right footed shot from outside the box is high and wide to the left.
Alim Ozturk (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Craig Curran (Ross County).
Attempt missed. Faycal Rherras (Heart of Midlothian) right footed shot from outside the box is close, but misses to the right.
Substitution, Heart of Midlothian. Jamie Walker replaces Don Cowie.
Alim Ozturk (Heart of Midlothian) wins a free kick in the attacking half.
Foul by Tony Dingwall (Ross County).
Second-placed Hearts were held to a goalless draw at home to Ross County in the Scottish Premiership.