Marco Di Vaio, whose numerous past clubs include Juventus, gave Bologna a first-half lead before Mirko Vucinic replied in the second half to give Juventus their fifth draw in their last six Serie A games.
Lowly Cesena, down to 10 men after only 21 minutes after a red card for Daniel Pudil, still managed a goalless draw at home to Catania who had Sergio Almiron dismissed in injury-time of a fiery encounter.
Sebastian Giovinco scored a late penalty to give Parma a 2-2 draw at home to Fiorentina in the evening's other match.
Juventus, using up their final game in hand, could have gone level with Milan if they had won but instead lie two points behind with 52 points from 26 games.
Juve remain unbeaten in Serie A and have played a further three games in the Italian Cup, also without losing.
Conte was forced to field Martin Caceres and Bonucci in a makeshift centre defence and they were quickly caught out when 35-year-old Di Vaio sprung the offside trap and steered his shot wide of Gianluigi Buffon in the 17th minute.
Juve enjoyed more possession but were toothless in attack until Andrea Pirlo slipped a ball through the Bologna defence and the much-criticised Vucinic brilliantly flicked the ball over the advancing Jean Francois Gillet.
The visitors should have won the game shortly afterwards when a Caceres header was saved by Gillet and Claudio Marchisio contrived to miss from the rebound with the goal at his mercy.
Conte was dismissed on the touch-line for insulting the fourth official.
Bonucci, already booked for a rugby tackle on Gaston Ramirez and having escaped a red card for a crude body check on the same player, left the referee with no option when he hacked down the Uruguayan again in the 84th minute.
Cesena, who started the day bottom of the table, moved level with Novara thanks to their stormy draw against Catania.
The Sea Horses were in trouble in the 20th minute when Pudil was booked for a foul, then sent off for protesting the original decision.
This led to chaotic scenes on the touchline where Mario Beretta, Cesena'a third coach of the season, and officials furiously remonstrated.
Tempers flared in injury-time when Almiron was sent off for allegedly head-butting Gianluca Comotto who fell over theatrically. An enraged Almiron tried to chase Comotto but was held back by team-mates.
Cesena have 17 points alongside Novara but Lecce are eight points clear of the pair in the final relegation place of 18th.
