NO! We have specifically engineered Brainscape to have minimal impact on your phone's battery and data usage.
Brainscape only downloads the flashcard content to your device upon initial usage, or whenever there has been an edit from another device. When you continue to use Brainscape, all flashcard data is being retrieved from the "local" database on your device, so we don't need to make any continual data-heavy calls to the server.
We do make occasional tiny data calls to our server to keep your study mastery data in sync between devices, but these are very "data-light". (A few bytes with "small" numerical data like Card ID and User ID.) So Brainscape should have a very minimal impact on your phone's data plan or your phone's battery consumption. (See also: Do I need a WiFi or 4G/LTE connection?)
We also do not use or share your Location data, which is often a big reason why other apps may be draining your battery.