sziklassy is the only one that really understood this question. I believe he's asking which app will use the least amount of battery, because he plans on it running in the background via Backgrounder, like we had done with Agile Messenger on 1.x.x.
Therefore, BeeJive's feature to keep you logged in is irrelevant, and has nothing to do with battery life while it's running.
This is an interesting question, and although sziklassy is on the right path by looking to how much data the app is processing, there are other factors, like the code itself, which will determine battery life of an app. I would assume the quickest and least flashy app would use the least amount of battery, perhaps MobileChat. Although an above poster characterized it as an epic fail, the developers are currently rebuilding the servers from the ground up to fix all problems, and are optimistic of success within a week or so. The app seems very simple and probably doesn't process a lot of information, thus leading me to believe it might have the least affect on battery life. Look to when the servers are all fixed, and keep an eye on their blog:
twenty08