Mourinho was a guest of Dortmund chief executive Hans-Joachim Watzke, whose team was frustrated for long periods by the home side.
Hertha might have gone ahead in the first half when defender John Anthony Brooks headed narrowly over from a corner.
Dortmund sent on attacking midfielder Christian Pulisic for the 17-year-old American’s second Bundesliga appearance, but needed a last-ditch tackle from Sokratis on Alexander Baumjohann to safeguard the draw in second-half stoppage time.
"Despite the lack of goals it was a good game," said Hertha coach Pal Dardai, whose side remained third, ten points behind Dortmund. "We could have earned more from it because we had more chances to win."
Schalke defeated Wolfsburg 3-0 to move fourth, two points behind Hertha, thanks to goals from Klaas Jan Huntelaar, Johannes Geis and Alessandro Schoepf.
Huntelaar struck a low shot to the far corner from around 25 meters (yards) in the 24th minute, Geis placed his free kick just inside the left post the 35th, and Schoepf came on for his second league appearance and completed the scoring with three minutes remaining.
"It could have been 3-0 or 4-0 after the first half. Then the second half was damage-limitation. It was too little today," said Wolfsburg coach Dieter Hecking, whose side is in crisis after its seventh league game without a win.
"You can see that it’s difficult for us in all areas at the moment. Schalke used that today," said Wolfsburg midfielder Julian Draxler, back at his home club for the first time since switching between the sides.
Elsewhere, Stuttgart won at Eintracht Frankfurt 4-2 to move five points clear of the relegation zone, Ingolstadt came from behind to defeat Augsburg 2-1, and Mainz won 1-0 at bottom side Hannover after Jairo Samperio took advantage of poor defending to score.
Bayern Munich visited Bayer Leverkusen in the late game. Bayern had the chance to go 10 points clear.
Both Stuttgart and Frankfurt finished with 10 men.
Stuttgart captain Christian Gentner opened the scoring and Daniel Didavi converted acrobatically before the break for the visitors.
Frankfurt captain Alexander Meier pulled one back before Gentner crossed for Georg Niedermeier to make it 3-1 in the 65th.
Didavi’s sending off shortly afterward gave the home side hope, but Frankfurt’s Carlos Zambrano followed him off in the 75th after giving away a penalty that Filip Kostic converted.