Team captain Smith and Warner were issued with 12-month bans for their roles in the incident, while Bancroft has been suspended for nine months.
Cricket Australia has revealed that Warner came up with the plan and demonstrated to Bancroft how to deliberately tamper with the ball using sandpaper during the second Test against South Africa in Cape Town.
Video footage showed Bancroft pulling out an object from his pocket before placing the piece of sandpaper down the front of his trousers.
Lehmann apologised to the "Australian public" while speaking to reporters and insisted that he had no knowledge of the plan, adding that he was "embarrassed" and "disappointed" with what occurred.
The coach, who has been in charge of the Aussies since 2013, condemned the trio but added: "These are young men and I hope people will give them a second chance."
While addressing the fact that Smith will be out of the game for a year and will be unable to captain his nation for two years, Lehmann claimed that "we've lost a great young man who made a mistake".
The 48-year-old added that the team and coaches "need to work to bring the respect back from the fans".
Cricket Australia ruled that Warner will not be considered for any leadership roles in the future.