The short answer is yes they do.
The longer answer is "sometimes". Tour members have Sponsor Value obligations that, depending on a players FedEx Cup standing and hometown popularity, can compel him to participate in a pro am, attend a sponsor or volunteer dinner, film a promotional spot, or interact with an events designated charities. Players have a couple of annual excused absences, but other than that it's mandatory. Many of these Sponsor Value obligations come with a media component. The LPGA is even more proactive in educating their players on the need to promote themselves and their Tour.
As far as the post rounders that Naomi Osaka was declining - Tour players are independent contractors. So on a day to day basis they can pick and choose who they say what to. Overall I guess there is a technical obligation. Although there is no reason to enforce it when a player has already decided they don't want to talk. Otherwise you'd just end up with a Marshon Lynch situation where a player is just saying "no comment" over and over again. But the media areas for the quick hit post rounders are usually directly in the players route from 18th green to scoring area to locker room. So if a player doesn't want to talk after a bad round he's still going to have to walk right past the media.
There IS an obligation in the player code of conduct section that prohibits them from making public comments "unreasonably attacking or disparaging" tournaments, sponsors, fellow members or the Tour itself. Penalties would fall under Conduct Unbecoming A Professional.
The paperwork you sign when you enter the Masters or PGA Championship has language compelling some media interaction. I haven't read the USGA rights release in a while, but I believe they require media cooperation if you want to collect all of your winnings.
Michael