Burnley's frustrating afternoon climaxed seven minutes from time when Chris Eagles was shown a straight red card by referee Mike Oliver for lashing out at an opponent. This capped off an afternoon to forget for Eagles as Burnley's £1million close-season signing had picked up an earlier yellow card.
Chances were few and far between throughout the afternoon but ironically it was Eagles who came as close as anybody to breaking the deadlock. The former Manchester United player was denied by visiting goalkeeper Graham Stack, who produced a fine save to tip over a free-kick from Eagles.
The resultant corner from Wade Elliott was headed over the crossbar by Burnley captain Steven Caldwell.
A quick throw-in from Elliott saw Chris McCann unleash a left-foot volley which Stack did well to keep out.
Argyle's Jason Puncheon did his best to produce a goal threat, setting up central defender Marcel Seip, but he pulled his shot wide.
Burnley goalkeeper Brian Jensen pushed away a free-kick from Steven MacLean while, at the other end, an Eagles free-kick was headed into the side-netting by Caldwell.
The visitors should have gone in front in first-half stoppage time as Puncheon found debutant on-loan striker Paul Gallagher but he shot inches wide.
Burnley manager Owen Coyle introduced Robby Blake for Kevin McDonald at the interval, but this did little to change the pattern of the game during the second half, as both teams struggled to create a meaningful opportunity.
A good piece of defending from Burnley's new signing Russell Anderson prevented MacLean from getting in a strike on goal and both managers then rang the changes in an attempt to nick all three points.
After the late dismissal of Eagles, Plymouth substitute Craig Noone did have a chance but he headed over a cross by Karl Duguid.
But it was Burnley that almost went in front deep into stoppage time as a blistering free-kick from substitute Joey Gudjonsson was tipped over by Stack and from the resultant corner McCann headed wide.