I use iCloud Photo Library myself, but I get it that people pay for other services and don't want to pay Apple's high prices, too. Google Photos, though, has full support for backing up Live Photos. See
http://www.theverge.com/2016/3/7/111...os-ios-support
I can't find any other stories about competing services (Dropbox, Flickr, etc.) supporting Live Photos in particular.
To be fair to OneDrive, I used the sharing intent to store the live photo there - I did not turn on photo backup. Since my iCloud Library has 15,000 photos, I figured it would take forever before OneDrive would have backed up the photo. Maybe backups are different from sharing a photo to the service?