In which country is unmarried cohabitation not legal?

Depends on your definition of marriage. Various places with common law automatically make you effectively married as far as family law and tax law are concerned, and insofar as people involve the courts.

After 1 year of cohabitation in Canada, couples are federally common law spouses for tax purposes. Provincially it depends, but after 2 years in some provinces you're technically and automatically spouses for family law purposes, which gives someone the same legal rights as an explicitly married couple in terms of asset division and parental obligations following a breakup (which is outdated and insane in many cases outside of having children, but whatever).

Iran, also in some US states like North Carolina and Missisipi (just not enforced).

Today, Iran. And even parts of Europe as late as the 90s

Burundi, since 2018.