I have that same app installed and i don't have that problem, maybe is still running in the background and that way keeps refreshing the tile with weather updates... hence more data consumption. Try to disable it in the battery saver in WP8.1 / Background task in WP8.
I found that Accuweather is running in the background. Sorry for the dumb question but it states that I have to turn off the battery saver to turn disallow it from running in the background?
Yes, if the app is running in the background most likely is still active and keeping refreshing the tile - app with weather updates (Eating mobile data) if it's turn off that would stop that process and only updating once you enter the app.