Sixth-placed Juve, who have 15 games left, have now won just once in eight league matches and rarely looked like taking three points in Tuscany.
Alberto Zaccheroni, in his second game as Juve coach having replaced the sacked Ciro Ferrara, changed tactics by opting for three at the back but the old defensive frailties remained.
Livorno's Antonio Filippini was unmarked when he nodded in Luigi Vitale's inviting cross midway through the first half.
Nicola Legrottaglie, only playing because of the switch to three centre backs, equalised after 42 minutes when he headed Diego's freekick past shaky new Livorno goalkeeper Rubinho.
"I'm not happy. We wanted to win," Italy's Legrottaglie told Sky television.
"It's a shame. We could have done more but it's a draw and bit by bit we are working on it."
Fabio Cannavaro added to Juve's long injury list by limping off with a thigh problem before Felipe Melo was sent off for two bookings late on.
Palermo in fifth moved two points above Juve and one behind fourth-placed Napoli after Simplicio blasted in an 87th minute winner to seize a 2-1 victory over Parma.
The Brazilian, who almost left for Inter Milan on Monday's transfer deadline day, fired in from 12 metres after Fabrizio Miccoli's clever pass.
Uruguay striker Edinson Cavani headed in Miccoli's free-kick to hand the hosts the lead in the second half before Jonathan Biabiany was allowed to equalise despite Cavani lying injured.
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