Have the same issue on a rooted installation on an Xperia ZL, lollipop. Neither the portal is able to contact the device, nor the device's app is able to connect.
I tried the "fix connection problems" wizard in avast app on my device, that too is not able to connect to avast server. the app says "The message sent from our servers to your device has not been received."
sent a feedback with a log. hoping for some resolution.
Totally clueless, as to why it stopped working all of a sudden. Avast Anti Theft is, as of now, "useless".