The hosts had the better of the first half, before Schalke gained the initiative to little effect after half-time.
Hapoel substitute Yossi Shivhon injected some much-needed life into a hitherto drab encounter and forced the best save of the night from Manuel Neuer, but neither team could find a way through.
Hapoel's Eran Zahani was booked for fouling Peer Kluge in an otherwise quiet opening to the game.
Kluge saw a shot blocked by Walid Badir as Schalke broke following a Hapoel corner before Toto Tamuz drew the first save of the night from Neuer, with 18 minutes having elapsed before the Schalke keeper parried away his 25-yard drive.
The hosts were looking the brighter, though, and striker Ben Sahar warmed Neuer's palms with a left-footed drive from the left of the penalty area.
Salim Toama was next to threaten but curled his free-kick over after Ivan Rakitic was booked for handball.
Hapoel keeper Vincent Enyeama was finally called into action, parrying after Sergio Escudero met Jefferson Farfan's cross, but Zahavi then drew another sharp save from Neuer.
Enyeama denied Escudero once more after Raul rolled a free-kick into his fellow Spaniard's path, but the visitors had Jose Manuel Jurado and Klaas-Jan Huntelaar booked either side of that incident.
Badir followed after a foul on Jurado, with Raul's header from Rakitic's free-kick saved unconvincingly by Enyeama.
The Nigeria international then tipped over Jurado's drive to end a scoreless half.
Christoph Moritz replaced Kluge at the break before Benedikt Howedes became the third Schalke player to handle and earn a booking.
Farfan flashed a shot wide and his efforts were soon mirrored by Rakitic, while Ha'adumim - the Reds - turned to midfielder Shivhon for an overdue spark, withdrawing Toama.
And the new man made an immediate impact, meeting Shay Abutbul's corner with a header at the near post and forcing a superb save from Germany international Neuer.
Shivhon headed wide shortly afterwards before going close once more from 20 yards, and Schalke then introduced a lively substitute of their own.
Ciprian Deac replaced Farfan and quickly saw a shot blocked by Dani Bondarv before delivering a corner which was not properly cleared, Christoph Metzelder eventually blazing over.
Another replacement, Goncalves Edu, headed wide as the match meandered to a goalless conclusion.