The visitors raced into an early two-goal lead with headers from Sylvain Legwinski and Alan Lee.
Legwinski opened the scoring on seven minutes with a powerful near post header past a flat-footed Burnley defence.
The Tractor Boys increased their lead two minutes later as Robbie Blake lost possession and Billy Clarke crossed for Lee to send a looping header beyond keeper Gabor Kiraly.
The Clarets pulled a goal back just two minutes later as Blake fed Wade Elliott and he crossed to the far post for Kyle Lafferty to head past Ipswich goalkeeper Neil Alexander.
Ipswich came close to extending their lead when Owen Garvan's corner found the head of Jon Walters only for Kiraly to tip the ball over the crossbar.
Tommy Miller went close to restoring Town's two-goal advantage only to shoot against the legs of Kiraly.
Burnley equalised against the run of play in bizarre fashion as goalkeeper Alexander raced out of his penalty area to clear an Alan Mahon throughball but only succeeded in smashing the ball against the outstretched legs of Burnley striker Andy Gray.
Alexander produced the game's best save when he dived to his left to keep out a header from Lafferty after a teasing cross by Stephen Jordan.
With 15 minutes remaining Garvan had a chance to grab the winner when he rounded Kiraly but his shot lacked power and was cleared by the home defence.
That almost proved to be an expensive miss as the lively Blake shot wide at the other end.
Lafferty produced a great piece of skill to jink between two defenders but his shot was scrambled away by Alexander.
Then came the goal that never was as Ipswich were denied what would have been a deserved three points.