There are two different things here. One, apps that run in the background are NOT using any battery power. Do a search here on "task killers" and you will find plenty of info on that.
Second, and somewhat contrary to my previous statement, some apps DO run in the background. Apps like music player (when you are paying music), naviagtion (when you are in nav mode), and apps like instant messengers (when they are signed in, so you can keep in touch all the time). That being said, I have yet to see a messaging app that doesnt let you sign out when you are not using it. I havent used Yahoo, but I'm sure there is an option to sign out and STAY signed out until you want to use it.