OverDrive for iOS (iPhone/iPad/iPod touch) does work with an Apple VGA or Digital AV adapter, but only on newer devices. Older iOS devices only output picture slideshows and video content from the "Video" app through Apple VGA and Digital AV adapters. As a result, VGA and Digital AV adapters cannot display streaming videos through OverDrive's app on older iOS devices.
Remember to check the digital rights for a video before streaming it in a public setting. For more information about video permissions and where you can find them on your library's OverDrive website, visit this help article.