The Adelaide Advertiser on Monday published a video on its website showing a man who appeared to be Mr Speirs sniffing an unidentified substance on a grey plate through a rolled up $5 note.
On Wednesday, Mr Speirs said the last few days had been "exceptionally distressing for me and my loved ones".
"I believe this is a deepfake or an elaborate hoax," he said in a statement.
"I have always put South Australians first, but now I must take some time to focus on my own health and wellbeing, as any other person should.
"I have consulted with medical professionals in the last 24 hours and have been advised that I should take time away from work to concentrate on my health."
Earlier, senior Labor government MP Tom Koutsantonis called on Mr Speirs to front parliament on Wednesday, after being absent from the house for 76 days.
"In that time, he has stood down from the Liberal Party leadership, attended a wedding in Scotland, repeatedly criticised his successor Vincent Tarzia and the South Australian Liberal Party organisation in general and dismissed troubling vision appearing to depict him snorting a white powder as a deepfake or elaborate hoax," Mr Koutsantonis said.
He added that Mr Speirs was adamant he had done nothing wrong, so it was time for him to return to the work he was elected to do.
"Vincent Tarzia must show leadership and tell Mr Speirs to turn up to work today," he said.
On Tuesday, Mr Tarzia said he had spoken to Mr Speirs and "he's assured me that he's of the belief that the footage is that of a deepfake".
"So my suggestion to David is that if he believes that that is the case, that he makes sure that he reports that to the authorities and I think he's taken that on board," he said.
The Adelaide Advertiser alleges the video was recorded at 4.12am on June 30, when Mr Speirs was still party leader, in the kitchen of his Kingston Park home in suburban Adelaide
In his statement, Mr Speirs said he left the leadership of the party on August 8 "as I had run out of energy and the feeling of burnout has been compounded in recent days".
"As a consequence, my emotional health and wellbeing has been significantly impacted, leaving me feeling exceptionally vulnerable and at risk," he said.
I ... am now on medical leave, but will be regularly consulting medical professionals."
He said he did not plan on making further statements regarding the video and had engaged legal counsel "to assist me to navigate this difficult matter".
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