We're technically using it in production for my site, www.bugdedupe.com, but we're still in beta, so we haven't experienced heavy usage.
We've been using Kubernetes to deploy Docker containers on Google Container Engine, and while there have been a few issues due to Docker/Kubernetes (namely, getting the containers to expose localhost to each other, and to expose themselves to the world), the issues that we've had so far have been issues that we would have been bit by eventually. Namely, if we hadn't been forced to deal with the issues now, we would have been screwed later. There have been some weird bugs due to the internal environment that containers use (we had extremely slow DNS lookups that caused our request times to shoot up to 9s each). These issues have been transient though, so it's not clear that it's Dockers fault, or if we're making mistakes in our code.
Docker has made our deployment much easier. You just build & push your container, and you instantly have a versioned deployable instance of your code. Kubernetes makes it extremely easy to rollout or rollback containers, so I have nothing but good things to say about containers.