When I installed the iOS 9 beta on my iPhone6 Plus it began to drain the battery very quickly and feels very warm all the time. Does anyone else have this problem?

Yes, as with almost each beta each year, battery life goes down drastically until closer to the final release. Both the watch and phone would easily get through a day where is in iOS 9 and WatchOS 2 barely makes it half a day. This is common and is why the betas are currently only available for dev testing purposes and not meant for using on your daily phone or watch. The final release should be the same or better battery life than iOS 8 and Watch 1.0.