How do I stop the rubber sweep on my front door from squeaking as it moves against the floor?
It's the same concept are wiper blades on a car. The rubber is either too far gone and need to be replaced or you can rejuvenate it by wetting it. Rubber, when exposed to the elements, will slowly dry out and lose its oils and become, well, dry. It can become brittle, squeak, and streak. If you can save it, then all you have to do it wet it and let it soak up some liquid just like your wipers working better after a few passes in the rains. You could use water, but it won't work well or for as long. Something like gun oil or any petroleum based oil would work well; however, it's also the first to seep out and cause oil streaks on your floor.

So how do you get a result you're happy with. 
If you can rub it while dry and it makes your hand dirty or crumbles, it's probably a goner. Otherwise, if you can find an oil you can stand the smell of (because you'll smell it for a while), soak the rubber in it for a few hours then let dry for another few hours. After that, rinse it thoroughly with hot water to get the extra oil sitting near the surface out.