The material posted to Sydney's Lakemba Mosque also singled out Middle Eastern communities, Aboriginal people and left-wing politicians, putting locals on high alert.
After forensic testing on the letter, received on Thursday, counter-terrorism officers on Saturday raided a home at Burwood, in Sydney's inner west.
Investigators seized more than 100 letters, stamps, notepads and several electronic devices.
A 70-year-old man was charged with three counts of sending a document threatening death or grievous bodily harm and is due to appear in court on Sunday.
The Lebanese Muslim Association said the timing of the threat, which coincided with Australia Day, was concerning.
"Australia Day should unite Australians of all races, faiths and cultures, and not be weaponised as a tool of concealed racism," Lebanese Muslim Association secretary Gamel Kheir said.
The receipt of the letter came more than a month after the Bondi terror attack, in which 15 people were killed by two Islamic State-inspired gunmen.
NSW Multiculturalism Minister Steve Kamper said the incitement of violence on Australia Day was "abhorrent", urging people to celebrate what united them and not let hatred divide.
In a second incident, a man handing out religious flyers escaped serious injury after being assaulted in what police called a religiously motivated attack.
The 52-year-old man was part of a group handing out flyers in Bankstown, in Sydney's southwest, on Saturday.
He allegedly got into an argument about the flyers with another man who had been walking past, police said.
The pair were not known to one another.
During the argument, the 51-year-old man was assaulted, although he did not need medical attention.
A 40-year-old man later approached officers at the local train station and was arrested.
He was charged with common assault and assault occasioning actual bodily harm, and granted conditional bail to face Bankstown Local Court in February.
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