How do I prevent followers from blocking doorways?

Most of the time, walking or sprinting into a follower will make them move. However, there have been occasions where no matter how much I ran into Lydia, she wouldn't budge. In that case, there is a simple solution. Just talk to her and select the "I want you to do something" option. You can then point at a location to tell her to move there, which should get her out of your way.

Just sprint into her, you will push her back; the default key on PC is Alt.

Alternatively, cast the Whirlwind Sprint...

For some reason, I've found that crouching (sneaking) consistently gets her out of my way. When I crouch, she crouches too and seems to shift her position just enough to let me get by.