Lee Martin, Alan Quinn and Carlos Edwards were all denied by Grant - though the visitors did have a strong case for a penalty waved away in the first half.
Ipswich are now unbeaten in six games but a lack of a clinical goalscorer continues to hinder their progress and they end the weekend four points adrift of safety.
Keane sprang a surprise in his selection with Owen Garvan returning in midfield.
The Irishman was expected to leave Portman Road on loan this week after being frozen out since August but was recalled after scoring a stunning goal for the reserves in midweek.
Full-back David Wright had the first effort on goal but his dipping 30-yard volley was easily held by Grant.
Garvan and Grant Leadbitter were dominating the midfield early on and they fashioned a chance for Tamas Priskin.
Leadbitter span a fine pass over the top where Priskin had peeled off his marker but the former Watford man delayed his shot and could only scuff into the hands of Grant.
Jermaine Johnson had the Owls' first effort but his shot was straight at Asmir Begovic in the Town goal before the visitors had a penalty shout turned away by referee Andy D'Urso.
Marcus Tudgay crossed from the left and Luke Varney flicked a header on - straight onto the hand of Damien Delaney.
The Irishman was certainly close to the ball and had little time to react but was fortunate to escape.
Martin blasted in a powerful strike which drew a spectacular save from Grant as the game opened up, and Begovic was equally as impressive in palming away a volley from Tudgay.
Edwards fired over when well placed as Keane's side pushed for an opener in torrential rain.
Leadbitter fizzed an effort inches wide in a lively start to the second period before both sides began to struggle in the deteriorating conditions.
Former Wednesday midfielder Quinn came on for Ipswich before Garvan tested Grant with a powerful volley from 15 yards.
Quinn should have broken the deadlock when he was released by a superb flick from Priskin but Grant produced a great save to keep his shot out.
And the weather almost gifted Ipswich the lead from the resulting corner when Grant was deceived by Leadbitter's wind-assisted corner and only just scrambled the ball clear from under his own crossbar.
Jon Stead came off the bench with 15 minutes left and forced Grant into yet another save with his first touch, with Priskin netting the rebound from an offside position.
Wednesday had a late rally but could not find the target as both sides were forced to settle for a share of the points.