US amateur exams

That's one of the test questions. If the FCC sends you postal mail to your address on file, and the mail is returned as undeliverable, you may lose your license. The relevant rule is Part 97, Section 97.23.

The FCC does its official correspondence primarily by US Postal Mail, and the US Postal Mail address is the address which is required to be on file for all licensees.

The ARRL can and does correspond largely by e-mail, but the ARRL isn't a government agency, and doesn't regulate licensees (they are one of the Volunteer Examiner Coordinator organizations, so they can do some administrative functions regarding license exams, though). You can be a licensed ham in the US without every being required to correspond with the ARRL, and failure to correspond with the ARRL will not result in any loss of license or loss of privileges.

I've only been licensed about four years, and in that time, the only thing the FCC has sent me is a copy of my license itself, right after I passed the test. But these days, they're not even sending that much. So the chances of them sending much mail, under normal circumstances, are pretty slim. But they may send you mail at any time, and you need to provide them with an address that will get the mail delivered to you. I'd guess the most likely circumstances to cause them to send you mail would be if they had a question or concern about your station operation, such as a complaint, a suspected violation, or something like that.