In a low-key game, played out in front of an empty Kop, the Championship side failed to break down well-drilled opponents, meaning a replay at Kennilworth Road a week on Tuesday.
The Owls made just one change, goalkeeper Keiren Westwood was rested, meaning a recall for Cameron Dawson.
That meant all 10 outfield players who started the 1-1 draw with Birmingham City on New Year's Day played.
And the hosts could have been ahead in the second minute, Barry Bannan's cross picking out Steven Fletcher at the back post, but the striker's header just cleared the crossbar.
The same Scotland combination created another chance for the Owls on 23 minutes, but Fletcher was off target from Bannan's corner.
Wednesday were forced into an early change when midfielder Sam Hutchinson departed, replaced by Joey Pelupessy on 25 minutes.
League One side Luton were putting on a good show, looking dangerous on the counter attack, but Adam Reach went close to putting the Owls 1-0 in front. The midfielder - who has banked some spectacular goals this season - juggled with the ball, before volleying just beyond goalkeeper James Shae's right-hand post.
Dawson denied Luke Berry's long-range effort, while at the other end Marco Matias saw his shot - after a short free-kick routine - flicked over by Shea.
The visiting goalkeeper did well to palm away Matias's fierce drive 10 minutes after half-time, as Wednesday tried to build up the pressure.
Wednesday brought on strikers Sam Winnall and Atdhe Nuhiu to try and force a goal, but the closest they came was a Bannan effort which was comfortable for Shea at his near post.
Elliott Lee almost snatched a late winner for Luton, but his shot fizzed just wide, while Liam Palmer failed to find a team-mate as he broke into the box.
The game finished with an injured Bannan taking treatment off the pitch on a frustrating afternoon for the Owls.