The Gunners got off to a great start at Turf Moor as Pierre-Emerick Aubameyang gave them the lead after just six minutes.
But the Clarets equalised before the break in bizarre fashion after Granit Xhaka’s attempted cross-field pass struck Chris Wood in the chest and ended up in the back of the net.
In the second half Arsenal had a strong penalty appeal turned down and also had one which was awarded overturned by VAR, while Dani Ceballos hit the post late on to cap off a frustrating day for Mikel Arteta’s side.
The result means they remain in tenth place in the table – and Lineker made reference to their disappointing season with a jokey tweet.
“Excellent point for @Arsenal. 38 points should guarantee their safety,” he wrote.
The game included yet more VAR controversy, and when asked in his replies what he thought of the two penalty incidents Lineker replied: “Yes, as always. Can’t stand it.”
Arteta was also far from impressed with some of the decision-making after full-time, and demanded an explanation for his side not being awarded a spot-kick when Burnley’s Eric Pieters appeared to deliberately handle the ball in the box.
“When you miss the chances we had, and you do not get decisions, it is complicated to win football games,” he said.
“If the first one is not a penalty honestly someone has to come to the training ground and explain what a handball is. It cannot be any clearer.”
Arsenal’s hopes of mounting a late challenge for a European qualification place appear to be fading fast, but they must next turn their attentions back to the Europa League.
They face Greek side Olympiacos in the first leg of their last-16 tie on Thursday ahead of the North London derby against Tottenham next weekend.