Substitute Moore, on for Robby Blake, needed just six minutes to head Burnley's equaliser after Grimsby had failed to clear a corner from Glen Little.
But six minutes later the Irishman was being stretchered from the pitch with a hamstring strain that could keep him out for a month.
Grimsby had gone ahead three minutes from half time following a mistake from Burnley skipper Steve Davis whose indecision allowed Stuart Campbell to gain possession and fire a right-foot shot past Marlon Beresford from the edge of the box.
There were few chances in the second half although Grimsby might have won it at the death when player-manager Paul Groves hit a half chance straight into the grateful arms Beresford.
Burnley might have had a couple of penalties but there was little in this performance to suggest a revival might be on the cards.
Clarets manager Stan Ternent said: "We were poor in the first half and looked a little lethargic at the end and we clearly have too many players below form.
"But there are still 27 points to play for and we have a squad more than capable of putting a strong run together.
"It's a tall order but while ever it's mathematically possible we've got to keep fighting for it."
Groves said: "It was a point we would have settled for beforehand and had we had a little more quality and patience about our play then we might have won it."
Grimsby have slipped into the bottom three and they suffered a further setback when Groves revealed afterwards that midfielder Stacy Coldicott had suffered a broken leg after a challenge with Davis.