App Store asks me for a password, but won't let me change the Apple ID I'm taking over someone elses job and am using the computer he was using. In the App Store there are some apps that ask me to enter a password and have an Apple Id already filled in but won't let me change it.

For example there's an update to Evernote. In the Apple App store when I click on "update" it prompts me for the password and has the e-mail address of the previous employee automatically filled in (and I can't change it). What can I do? Evernote itself is signed out, so there definitely wouldn't be a loss if I reinstall it.
It won't let you change it because the apps are tied to the account that purchased them.  See here for the legal details.

Basically, only the original purchaser can update.  If you'd like to be able to keep up-to-date with an application, you must have it tied to your account...by buying/downloading it yourself. If that's the case...


Delete the application


open Finder
select Applications folder in left pane
drag apps tied to the old account to trash

Ensure that the old user is signed out of the Mac App Store
Sign in under an account you control
Purchase/download to your heart's content