The heavy rain that arrived shortly after lunch left the ground saturated and, following a 5pm inspection, umpires Martin Bodenham and Steve O’Shaughnessy had little option but to abandon play for the day.
It left Leicestershire on 142 for three, still needing another 276 runs to avoid the follow on in their reply to Northants 567 for seven declared.
But with only one day remaining, Leicestershire will now have high hopes of saving the game.
They began the third day at 47 for two, with Burns on 15 and Ned Eckersley 19, and both batsmen rode their luck in the early overs with Andrew Hall causing all manner of problems with his accurate medium-paced cutters.
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But showing plenty of resilience, the third-wicket pair survived, with Burns particularly impressive.
The 23-year-old Australian has struggled for form since arriving a month ago to replace Ramnaresh Sarwan, away with the West Indies in the Champions Trophy, as Leicestershire’s overseas player.
In his previous six innings in the Championship and Yorkshire Bank 40, Burns had a top score of only 21. That was on his debut against Glamorgan last month.
But he gradually grew in confidence as his innings progressed and a glorious straight six off James Middlebrook put him in sight of his first half-century for the county. It eventually came off 81 balls with six fours plus the six.
Eckersley also looked solid, but having reached 43 he was trapped lbw on the back foot by Middlebrook to end a stand of 99 in 34 overs.
Leicestershire went into lunch on 133 for three but only three more overs were possible after the break before the rain swept in, with Burns on 67 and Matt Boyce on six.