The former Celtic manager was confronted by angry supporters as he trudged over to them at full-time in the Premier League contest.
Video footage has shown major discontent amongst those who travelled to the Vitality Stadium with Postecoglou facing a barrage of abuse.
WARNING: Very strong language
The Australian coach appears to exchange words with a number of supporters before appearing to point at the Spurs badge on his jacket.
As supporters continue to vent their frustrations, Postecoglou then quickly applauds fans before walking back across the pitch.
The loss leaves Tottenham in 10th place in the Premier League with six wins, two draws and six losses in their 14 outings so far.
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Quizzed on the confrontation after the match, Postecoglou told Sky Sports: "I wasn't having a word. They are pretty disappointed and rightly so.
"I got some pretty direct feedback as to how we are going, and that's fair enough."
Tottenham next face Chelsea as they look to end their four-match winless run. They drew with Fulham and Roma before the defeat to Bournemouth.
But Spurs did enjoy a thumping 4-0 win over Manchester City last month in their most recent victory.