Joe Cole's first-half effort set Villa on course for a much-needed three points as Paul Lambert's injury-hit side put in an encouraging display at Turf Moor, but Ings pegged them back with a last-gasp spot-kick.
Villa, who had claimed just two points from their previous eight league matches, carved through the Burnley defence at will during the first 25 minutes, with Gabriel Agbonlahor and Ciaran Clark going particularly close.
Cole gave the visitors a deserved lead in the 38th minute with an instinctive finish, but Burnley did improve after the break.
Sean Dyche's men were aiming to win a third consecutive Premier League match for the first time in their history and nearly drew level through Michael Kightly 10 minutes into the second period.
The hosts' persistence eventually paid off, though, as Jores Okore bundled Lukas Jutkiewicz to the floor inside the area and Ings buried the resulting penalty to steal a point with three minutes left.
Villa's preparations were hardly helped by Roy Keane's decision to relinquish his role as assistant manager on Friday, though the visitors did look threatening during the early exchanges.
In fact, Lambert's men went agonisingly close to taking the lead in the sixth minute as Okore flicked Ashley Westwood's corner towards goal, only to see Dean Marney clear from underneath the crossbar.
Villa continued to pile the pressure on and caused more problems from a corner a few moments later, with Marney again decisive as he blocked Clark's goal-bound header.
Burnley eventually created their first chance of the match in the 14th minute as Kightly cleverly flicked the ball over his own head and then shot right across the face of goal.
Agbonlahor really should have put Villa ahead shortly after, though, as the rapid forward linked well with Cole and then shot straight at Tom Heaton when one-on-one.
Villa eventually took a hard-earned lead seven minutes before the break; Cole poking inside the post after latching on to Andreas Weimann's well-weighted pass.
Burnley started the second half brightly and almost restored parity in the 56th minute as Brad Guzan denied Kightly from close range after the forward dribbled past a couple of defenders.
Villa remained a very potent threat on the counterattack, however, and Heaton was forced into action two minutes later to palm a Weimann effort around the post.
Burnley were finally rewarded for their late pressure as Okore hauled second-half substitute Jutkiewicz to the ground in the box and Ings confidently smashed home the penalty.
The hosts continued to pile men forward and twice they went close in stoppage time as Ings hit the post and then Jutkiewicz headed just wide, but ultimately they had to settle for a point.
