It was the Shakers’ first point in five weeks, while Dons extended their winless run to four games.
The visitors were stunned not to be awarded an early penalty when Efe Sodje appeared to bring down Daniel Powell just inside the box, the referee instead awarding a free-kick.
Dons easily had the better of the first half chances. Jabo Ibehre wasted a three-on-one break, pulling his shot tamely across the face of goal, while Powell twice shot straight at Bury goalkeeper Trevor Carson.
Carson then ensured it remained scoreless at the break, diving full stretch to deny Dean Bowditch.
Martin saved from Andy Bishop in the final minute, while Carson pulled off a super save to deny debutant Clinton Morrison an injury-time winner.