Smith went off injured after being struck on the neck by a fierce delivery from Jofra Archer on Day 4, though returned to bat in Australia's first innings at Lord's.
The 30-year-old was eventually dismissed on 92, as Australia made 250 all out, eight shy of England's first-innings total, with the hosts extending their lead to 104 by the close of play.
However, Smith did not return to the field at the start of day five and Cricket Australia (CA) confirmed the former captain will sit out the rest of the Test after he complained of a headache on Sunday.
A CA statement read: "Steve has been closely monitored by medical staff overnight and this morning reported that after sleeping well, he woke with 'a bit of a headache and a feeling of grogginess'.
"Steve reported that his left arm which was also struck during his innings yesterday was 'much better'.
"As part of the Cricket Australia concussion protocol, repeat concussion testing of Steve Smith was also performed this morning and demonstrated some deterioration from his testing which is consistent with the emergence of the symptoms he was reporting.
"On that basis Steve has been withdrawn from the match by team doctor Richard Saw."
Smith's absence will present a real blow to Australia, with the Test hanging in the balance, though Tim Paine's side have been allowed by the ICC to replace the 30-year-old with Marcus Labuschagne
The ICC introduced new rules prior to the Ashes for concussion replacements, though the incoming player, who will be allowed to bat and bowl, must be deemed a "like for like" replacement.
CA confirmed Smith – who will have a precautionary scan on his neck on Sunday – will continue to be monitored ahead of the third Test, which begins at Headingley on Thursday.