How important is the leave address on DA 31?

Approved Leave is granted and if you travel out of state and not on a recall notice then you have no fear. Your plans do change and when you have to submit the 20lbs of required forms with your Leave form the best advise it to always pick the farthest place to your relatives as your radius of travel and then always update the signed leave form with your corrected address attached to it when signing out and notify your Supervisors accordingly. Now if overseas travel that is another topic and does not apply to this discussion

I went camping along the Oregon coast when I was stationed at Ft. Lewis. My address was "Oregon coast", phone # CB channel x (before cell phones). Another time I spent a week on a house boat on Lake Powell. I put down the name and address of the emergency station, the name of the boat, and what channel we were going to be monitoring. The command just needs a way to get in contact with you if something happens. When I was on leave I didn't always stay in one place. Make sure they have a good way to contact you and then enjoy your leave.

It shouldn't be a big deal unless there is a recall order sent out and you don't comply in a timely fashion, or there is a disaster of some sort at the address you did provide and your chain of command gets worried enough to check on you but come up empty-handed. Or if there's a disaster at the address you didn't provide and you disappear off the radar long enough to concern someone.
Otherwise, your leave address is just unused information.