A locked gate like this on DNR land, left open usually means log or gravel trucks are running/in out. I'm pretty sure they're logging up there (I think you're talking about the Marsh Creek mainline). Time is money for these guys and they're not going to mess around opening/closing a gate when they're working. That's why you leave it open if you find it that way.
The sign that says "no-motor vehicle access" still applies, you risk getting a ticket; or worse getting locked in.