Everton's victory at Huddersfield means the Clarets are not mathematically assured of a top-seven finish, and a place in next season's Europa League, though they are six points clear of the Toffees with two games remaining, and have a far better goal difference.
Burnley, whose last European campaign was during the 1966-67 season, came closest to scoring at Turf Moor when Kevin Long hit a post in the first half, yet Brighton also picked up a valuable point in their battle to avoid the drop.
Sean Dyche's side started the afternoon knowing they needed other results to go their way even if they won, but they seemed intent on sorting their part of the equation.
Ashley Barnes and Chris Wood both came close to sweeping in Ashley Westwood's low ball, while Johann Berg Gudmundsson drove wide after a purposeful run.
The hosts were nearly caught out from a corner when an undetected Bruno volleyed wide, though it was a 24th-minute set piece at the other end that nearly produced the opener in a frenzied passage.
Bruno failed to clear Gudmundsson's free-kick and Mat Ryan was forced into two saves from Barnes and Jack Cork, eventually scooping the ball away on his own line to James Tarkowski, who back-heeled to Long to hit a post from point-blank range.
The ball then popped up and appeared to strike Lewis Dunk on the arm before going behind for a corner, yet referee Roger East was not interested in awarding Burnley their first penalty in the Premier League for more than a year.
Ryan made another two saves, one that was straightforward from Barnes' swept attempt and another where he pushed out Gudmundsson's free-kick, before the Clarets appealed for a penalty again.
This time Long nodded on to Bruno's outstretched arm, though his proximity to the ball ensured he got the benefit of the doubt from East.
Burnley's lengthy wait for a penalty might have added to the frustration on the stroke of half-time when the hosts were angry about not receiving the ball back from a drop-ball situation following an injury to Gudmundsson.
Players from both sides shoved opponents as they made their way towards the tunnel, where news of Crystal Palace's lead at Leicester would have raised the hosts' spirits.
Pope smothered Pascal Gross' attempt at his near post shortly after the restart, while Aaron Lennon got in Wood's way as the New Zealander prepared to pull the trigger, but both managers recognised changes may be needed to break the deadlock.
Wood, seemingly injured in that attack, came off for Sam Vokes, who almost made an instant impact but just failed to collect Tarkowski's knockdown from a free-kick.
Seagulls boss Chris Hughton made a change in introducing Glenn Murray for the final 23 minutes, though seeing him pick up an early caution was not what his manager had in mind.
And neither side could conjure up a winner, although a point should be enough for them to achieve their respective goals of European qualification and safety.