Essential maintenance for my bike whilst on a long distance tour
This will depend on what kind of conditions you're riding in, what condition your bike is in, etc, but I think these are the basics:


Air up tires daily.
Lube chain. How often will depend on what kind of lube you use, which is one of those religious debates I prefer to avoid.
Check headset for looseness. This actually was an issue for me. Also check bearing cones (if you have loose bearings) on your hubs.
Check wheels for trueness, spokes for tension.
When you get a flat, make sure to remove the offending pointy thing from your tire. This can be harder than it sounds.
Check frequently (every time you stop) for slow leaks. I destroyed a tire prematurely because of a slow leak.
Check tightness of rack fixtures daily.


Naturally other things can go wrong, but I think these are the main things that you can rely on going wrong or needing attention.

And don't forget maintenance on your body. Be prepared for sunburn, saddle sores, hotspots on your feet, muscle aches, etc.