Town wasted their second great chance to claim sixth spot in four days after taking the lead through Paul Anderson in the first half and Paul Green 11 minutes after the interval.
But the plucky Cherries deservedly clawed them back on both occasions with a fantastic overhead kick by central defender Steve Cook and Matt Ritchie's cruelly deflected free kick late on.
Bournemouth are now three-points adrift of sixth while Ipswich are just a single point back with two games to play.
And, while Town boss Mick McCarthy vowed to carry on the fight, Howe has virtually conceded any hope of a top-six finish 12 months after claiming promotion from League One.
He said: "We always knew it would be difficult this season. We needed an incredible run and we did that initially, but it was difficult to maintain.
"But if we don't make it this year, we will have plenty to build on as we have got a really good squad of players."
Despite missing out on a chance of ending the night in the coveted sixth position, McCarthy said: "Bournemouth are a good side and I think it helped that they had an extra day (after playing Friday rather than Saturday).
"I think it is a good point for us and a really good advert for the Championship. So I can't express enough how proud I am with the lads.
"They do their job every single week and put in so much effort for themselves, for me and their club. They really honour the shirt.
"I said all along the league is bonkers. Everybody desperately wants it and that is why it is harder to grasp. Everyone is playing for something but it is a tough old league.
"We have two games to go so we will see. We might win both and that might be enough or we may not."
Ipswich took an undeserved lead in the 36th minute when Cole Skuse broke clear and centred for Anderson to fire home from close range.
But they were unable to hold onto the advantage at the interval thanks to Cook's outstanding equaliser.
The pattern continued in the second half when Green headed home Aaron Cresswell's free kick to restore Ipswich's lead, but they were once again pegged back thanks to Ritchie's heavily deflected set-piece.
Both sides could have claimed a vital win at the end with Tyrone Mings' header missing the far post for Ipswich and home goalkeeper Dean Gerken denying Yann Kermorgant and Lewis Grabban.
Howe said: "From my perspective, I thought we were the better team, created the better chances and played the better football."
Of Cook's brilliant equaliser, the Cherries boss added: "I have never seen him do that in training or in a match - I didn't know he was capable of it."