The Bulldogs' general manager of football Phil Gould announced the star player's decision at a snap press conference on Tuesday morning after meeting with club bosses.
The winger will miss at least Sunday's elimination final against Manly, the Dogs' first finals appearance since 2016.
Addo-Carr has told Canterbury officials he did not consume illicit drugs last week, but stood himself down after it was explained to him by Gould that his presence in the team would likely create a media circus.
"We accept Josh's belief of his innocence and we will support him through the process," Gould said.
"Josh has also engaged lawyers in his own personal defence, but that is a matter for him to report on later.
"But as far as the short term is concerned, he won't be considered for selection this week in the best interests of the club."
Addo-Carr was pulled over on Friday night while driving in the Sydney suburb of Wentworth Point and allegedly returned a positive test.
He was then tested a second time, with the sample sent away for further analysis.
Addo-Carr told Bulldogs officials on Saturday morning the test was inconclusive, but he was in the clear as he was able to drive after the test.
It was only when the club received phone calls about the matter on Monday they discovered that the Australia Test star had allegedly returned a positive sample.
Gould said he accepted Addo-Carr did not intentionally mislead the club.
But he indicated on Tuesday that it was possible Addo-Carr would be playing this weekend if he had been clearer in the initial details on Saturday morning.
"Under the CBA rules, we have no reason to stand him down, nor has the league until that last sample comes back positive," Gould said.
"So with that in keeping, there would be no reason for us to stand him down and go through the normal process.
"Josh has at all times defended his innocence, and said he didn't take this and didn't do it.
"Did he lie to us? I'm convinced in Josh's mind that he didn't think we'd ever hear about this.
"So he probably didn't tell me the whole truth of the first positive or what it read. He put that down to being inconclusive.
"So he hasn't handled that part well.
"But do I think he deliberately lied to mislead me? No. That's Josh."
Josh Addo-Carr is tackled by Manly's Lachlan Croker during his last match for Canterbury. (Dan Himbrechts/AAP PHOTOS)
Canterbury anticipate that it could take up to 10 weeks for the secondary sample to be returned.
Gould suggested that if it came back positive, Addo-Carr's future at the club would fall under the spotlight.
"That's something we will deal with there," Gould added.
"There are sanctions to that and precedence for that.
"If it came back positive, Josh would have to explain how it got in his system.
"Because he is adamant, and I mean vehemently adamant, that he has not taken the drugs."
Jeral Skelton appears the likeliest man to come in on the wing for the Bulldogs this weekend, provided he can recover from a minor leg injury as expected.
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