Both teams needed all three points for very differing reasons, with Chelsea trying to keep the pressure on title rivals Manchester City and Liverpool, and Norwich looking to move out of the relegation zone.
Chelsea huffed and puffed but could not break down a resolute Norwich defence, for whom Michael Turner excelled.
Schurrle beat John Ruddy but hit the upright with a right-footed shot in the first half, while second-half sub Luiz smacked the crossbar with a fierce drive.
Both teams had strong shouts for a penalty turned down by referee Neil Swarbrick, but in truth neither side showed the quality or desire to earn three crucial points.
The draw leaves Chelsea, who visit relegated Cardiff next weekend, one point behind leaders City and second-placed Liverpool, who have two games in hand, while Norwich’s relegation could be completed on Wednesday.
A draw between Sunderland and West Brom on Wednesday would all-but seal the Norfolk side’s fate as they have such an inferior goal difference that a win at home to Arsenal next Sunday would be immaterial.
Eden Hazard collected the Blues supporters’ player of the year award on the pitch before kick-off, wearing a tracksuit after dropping to the bench as one of five changes following the midweek loss to Atletico Madrid.
The Belgium forward, who has been continually linked with Paris Saint-Germain, was criticised by Jose Mourinho in the wake of the Blues’ elimination as “not the kind of player to sacrifice himself for the team”.
His place went to Schurrle, who shot straight at John Ruddy after Turner gifted the German the ball.
Turner then stood his ground to bodycheck Demba Ba as the striker surged into the box.
Norwich might have had a penalty in the 11th minute when Bradley Johnson fed Martin Olsson, with Ashley Cole and John Terry each diving in. The Blues captain clattered Olsson, but Swarbrick dismissed appeals.
Shorn of the beard he sported in recent weeks, Cole showed more attacking zip and saw one cross towards Ba intercepted by Ruddy.
But Chelsea lacked the incisiveness of Cole’s razor, giving Norwich hope.
Terry headed straight at Ruddy from a corner by Schurrle, who then hit the post after Nemanja Matic dispossessed Robert Snodgrass and brilliantly found his team-mate down the left.
Schurrle cut in on to his favoured right foot and beat Ruddy with his shot across goal, but the ball bounced back off the foot of the upright and Norwich scrambled clear.
Lampard was replaced at half-time along with the equally ineffective Mohamed Salah and their replacements fashioned a chance for Chelsea.
Hazard dazzled in the corner, fed Branislav Ivanovic, who in turn found Luiz to strike a rasping first-time effort against the bar.
Chelsea had a penalty appeal waved away when Alexander Tettey lunged at Schurrle inside the box, and another when Hazard ran into Bennett, who did not make contact with the man or ball, which went to Terry in an offside position.
Ivanovic was booked for his protests and Mourinho made a brief exit down the tunnel before returning to tell fourth official Michael Oliver his opinion.
The Portuguese has long advocated television replays and was perhaps making use of a screen.
Nathan Redmond was introduced by Neil Adams and surged at Chelsea’s defence, freeing Snodgrass whose shot was blocked by Gary Cahill.
The chance roused Norwich and Chelsea, who threw on Fernando Torres for Matic in a bid to find a winner.
Hazard was denied by Ruddy and Norwich found space as Chelsea pushed forwards, but all too often crucial passes went awry.
Bennett’s last-ditch tackle denied Torres in stoppage time as Chelsea failed to find a late winner.