The tour, scheduled for July, with two test matches against the All Blacks, will now not take place.
It has left New Zealand Rugby scrambling to save their July test window.
"After constructive consultations between our federations and health authorities, to our deep regret, the current conditions made it preferable to concentrate energies at home ahead of the autumn series," FIR President Marzio Innocenti said in a statement.
Italy has registered 122,694 deaths linked to COVID-19 since its outbreak emerged in February last year, the second-highest toll in Europe after Britain and seventh-highest in the world.
The country is still struggling to contain the virus, having reported 224 coronavirus-related deaths on Saturday against 207 the previous day, the health ministry said, while the daily tally of new infections fell to 10,176 from 10,554.
The Azzurri are currently due to host the All Blacks in a November international as well as taking on Argentina and Uruguay in the same series.
New Zealand, on the other hand, has registered 26 deaths from COVID-19 in total after the government enforced strict lockdowns and social distancing rules that helped to virtually eliminate the virus.
Reports in New Zealand suggest there are plans to pit the All Blacks against Fiji in back-to-back tests and add Samoa to the three-game window instead this summer.
The All Blacks have never played an entire home series against Pacific Islands opposition.
