I've noticed that when I'm not on my Wilds plot, if I start my furnace up with a load of ore, if I go offline, or to another world, the furnace stops and only restarts when I return to the Wilds plot.

Is this intentional? It's a bit of a bother, having to stand there while the furnace does it's business. Especially if I'm online, but in a different world, like Town or Mining.

As explained, this is part of Minecraft. It's how things work, everything; farms, furnaces, animals, redstone, etc. it all only works when you're near it. If you're not near the furnace it won't be working. This isn't going to be something we'll be looking to change either, things stopping when the land isn't loaded is done on purpose as to reduce stress on the server and client.

I didn't know, so now I know.

One thing though, I noticed that my neighbors' redstone devices were running. Was that because I was visiting his place and loading the chunk?

yes