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Medo was next to come close for Bolton with a powerful effort from the edge of the box, but it was always moving high and wide of the target.
David Jones, who joined Burnley on a free transfer in the summer, also tried his luck from outside the box after 26 minutes, but the former Manchester United midfielder's effort flashed past the post.
The tempo continued in the early stages of the second period with both sides committing numbers to attack, but neither could find that elusive pass in the final third.
Junior Stanislas twice tested Bogdan either side of the hour mark, but the Bolton goalkeeper was on hand to deny the former West Ham United winger.
Dean Marney was next to try his luck from distance for the home side, but the midfielder could only flash his effort high and wide of the crossbar as the opportunities continued to dry up.
Both sides made changes in the latter stages, with Craig Davies testing Heaton with a powerful effort outside the box after 82 minutes.
Ings created some space outside the box for Burnley moments later before firing an effort towards goal, but Bogdan was always comfortable in averting the danger.
Another long-range strike came from Medo later on, but the Bolton midfielder could not find the target as the two clubs opened their league season with a point apiece.