Midway through the first half, which was bereft of anything approaching a chance, the Killie striker took advantage of hesitancy from Accies' defender David van Zanten to race clear on goal.
However, the Hamilton goalkeeper kept his nerve to block Burchill's unconvincing strike and thereafter neither goal saw much action.
Second-bottom Killie, still looking for their first win under new boss Jimmy Calderwood, are now two points ahead of basement side Falkirk but the Bairns could come off the bottom with a win over Motherwell tomorrow.
Hamilton revealed after the match that they had rejected a bid from West Brom to sign midfielder James McArthur on a loan deal.
There was a late disruption to the Accies' line-up when skipper Alex Neil was injured in the warm-up and had to be replaced by David Elebert.
However, the home side started enthusiastically, forcing a couple of corners within the first few minutes but without troubling Killie keeper Cameron Bell.
In the eighth minute it was Rugby Park striker Kevin Kyle, at the other end, who flicked a Craig Bryson corner high over the bar from 14 yards in what was the visitors' first attack.
Any early optimism, though, appeared misleading as the match deteriorated by the minute, with mistake following mistake.
The game refused to improve with time as both side showed why they are stuck in the relegation battle.
But Kilmarnock had a great chance in the 25th minute as Van Zanten hesitated in defending a Mehdi Taouil clearance, allowing Burchill to nick the ball and race clear on goal.
The former Celtic striker only had Cerny to beat but the Accies keeper stood his ground and blocked Burchill's low drive for a corner which the home side defended.
Moments later, to add injury to insult, Burchill limped off to be replaced by Liam Kelly.
Neither goal was really threatened before half-time.
Kyle had a tame header saved by Cerny seven minutes from the break before Bell had just as easy a task in saving from Mickael Antoine-Curier's close-range drive.
McArthur's drive from 25 yards was again mopped up by Bell as the match spluttered all the way to the break.
The small crowd of 2,018 inside New Douglas Park looked forward to some improvement after the break but the early signs were not encouraging.
Accies boss Billy Reid made a double substitution 10 minutes into the second half, Jordan Kirkpatrick replacing Marco Paixao and Kyle Wilkie on for Elliot.
On the hour mark Killie captain Manuel Pascali powered in a header from a Bryson corner but Cerny was perfectly placed to gather.
A minute later Trent McClenahan came on for Richard Hastings as Hamilton boss Reid used his last option off the bench.
It looked increasingly like the first goal would be the winner, if a chance could be fashioned.
A brief spell of Accies pressure with 15 minutes remaining saw Antoine-Curier just fail to get on the end of Wilkie's cross before Van Zanten's drive was blocked by a clutch of Killie bodies inside the box.
The closing stages failed to deliver a grandstand finale and both sides settled for a point which neither really deserved.