The fixture in the North-East was regarded as a must-win for the Black Cats, but they could not break down a Clarets side who registered just their third away point of the season.
Throughout the remainder of the first half, the visitors had the better of the play with Scott Arfield sending a free kick just wide, but Sunderland remained a threat with Darron Gibson also just failing to hit the target.
Five minutes before the break, Burnley came close to opening the scoring, but Barnes poked just wide after Arfield had miscued his shot.
The Welsh forward had another opportunity two minutes later when he raced through on goal, only to find Pickford equal to his shot as Burnley's search for an away win continued.
In added-on time, Wahbi Khazri brought a save out of Heaton but neither side deserved to lose on Saturday afternoon as Sunderland failed to secure the win they required to give themselves renewed hope of remaining in the top flight.