To answer my own question, according to Wikipedia:

"Several states on the Eastern seaboard, the Southeast, and Gulf Coast (except Texas) have enforced vehicular laws since the early 1990s that require headlights to be switched on when windshield wipers are in use. This prompted the phasing in of DRLs in the affected states (from Maine to Florida including Louisiana, Mississippi, and Alabama)."

So it appears that DRL aren't required, but frequently standard equipment in states that require headlamps on if windshield wipers are on... Wikipedia does not list any states requiring use of headlamps all the time, though.