Non-executive director Brian Wilson will become interim chairman from 31 December until a permanent appointment is made.
Lawwell, who became Celtic chief executive for 18 years from 2003, told the club website it was “with sadness” that he was ending his three-year spell as chairman.
“I believe that my 18 years as chief executive and three years as chairman at the Club have shown my ability to meet and overcome challenges on many fronts, but abuse and threats from some sources have increased and are now intolerable,” he said.
“They have dismayed and alarmed my family. At this stage in my life, I don’t need this. I cannot accept this and so I leave the club I have loved all my life.
“The motivations and aims of these detractors can be investigated by others. I prefer to look back on my career at Celtic with deep gratitude and satisfaction.”
The announcement came hours before Nancy takes his squad to Tannadice to face Dundee United in Wednesday’s Premiership fixture and as Rodgers is poised to hold his first media conference as manager of Saudi Pro League club Al-Qadsiah.
“My time at the club has brought joy to someone whose memories as a Celtic fan include standing as a schoolboy on the terraces as the European Cup was paraded at Celtic Park on that magical night in 1967,” Lawwell said.
“I have subsequently watched some great European nights at Paradise and am proud that the club has played group-stage football in all but two of the last 21 years while also enjoying domestic dominance.
“My life at the club has included dramatic ups and downs in football, but they have also given me the experience, drive and desire, which I will take on to my next project, whatever and wherever that may be.”
Principal shareholder Dermot Desmond paid tribute to “all he has given” to Celtic over so many years.
“I am personally very sorry to lose Peter as our chairman, but I understand and respect his decision in light of recent events and what he has endured,” he said.
“As chief executive, Peter was the driving force of the club and, in his later role as chairman, he oversaw board affairs.
“Peter has witnessed 38 cup or title triumphs in his time at the club, a period of extraordinary domestic dominance for us as we have also competed in European football.”
Desmond also pointed out that Lawwell had overseen the building their training centre at Lennoxtown, the creation of the new facility at Barrowfield and the upgrading of Celtic Park.
Chief executive Michael Nicholson added that it “is a sad day for Celtic” and that “the abuse and threats” directed towards Lawwell and his family “are completely unacceptable”.
