Definitely disappear eventually.
I actually think a combination of options one and three should be in effect. I think option one should persist for one month, so if you don't log in for a month, your stuff is still there and nothing changes. However, after the month passes, then your beacon starts to require fuel. Maybe the beacon has two months of fuel, and if that runs out then it disappears. I also really love the idea that you can hand the keys over to another player or a guildmate to take over your beacon if you know you're going to be away from a computer for a long while. I think this is a must have option, that way the area is still being used and getting occupied.
I think the beacon's power should be able to stem off the natural growth of the natural world state if its in prime condition (the player has logged in <= 1 month ago). However, I think as it runs out of fuel in months 2-3, I think it should lose its ability to fight off natural growth, so that it starts to look like an abandoned place. Honestly, I think abandoned places are cool as heck, especially overgrown ones. Would love to see this. So the less fuel it has, the more the natural realm will take hold of it until, poof, beacon vanishes and the buildings are overgrown and finally melted back into the natural world state. This would also allow for a person or even a guild to come in and smash away the extra dirt and shrubbery that might take hold of the person's abandoned place, so they can preserve it until they finally take over it when the beacon vanishes. Other players shouldn't be able to fuel the beacon, unless the owning player gave them the right to.
My question is, if this were to happen, should everyone be able to see how much fuel the beacon has left (how much time it has left until it disappears)?