Although Joey Gudjonsson struck the top of the Wednesday crossbar with a 24th minute corner, both sides generally played with a lack of confidence.
Deon Burton should have put the Owls in front when a low cross from left-winger Chris Brunt found the Wednesday striker, but his carefully placed shot was turned behind for a corner by Wales international keeper Danny Coyne.
Burnley had strong claims for a penalty turned down after defender Lee Bullen appeared to handle the ball.
Coyne then made another fine save to deny Tommy Spur after Brunt's corner found him unmarked at the far post.
Coyne made it a hat-trick of first-half saves when he tipped a Brunt free-kick behind after John Harley was adjudged to have fouled Wade Small on the edge of the penalty area.
With half-time approaching, Wednesday keeper Mark Crossley was called into action to save a shot from Ade Akinbiyi on his near post.
It was a surprise when Burnley went in front as Akinbiyi's header sent Wade Elliott clear to score, but Wednesday equalised within three minutes as Brunt's corner found head the head of Burton.
Both sides had opportunities to win the game with Andy Gray and Elliott coming close for the home side, while at the other end Steve Watson drove inches wide and Coyne made another fine save to deny Small.
Small was then again involved in the action, seeing his shot deflect off John McGreal and strike the foot of the post.
Then it was the turn of Brunt to strike the woodwork with an attempt from long-range.
A draw was perhaps a fair result, with both sides still nervously casting an eye towards the foot of the table.