From what you've described, I don't think this is an HDMI issue at all. That you don't have an issue with live TV, for me, isn't strange but one of the glaring clues...
All the functions you've referred to use the internet for connectivity, and TiVo's have this connection inbuilt. It's quite a common thing for the onboard 'net connection to fail and users to not realise.
On the front-left of your TiVo are a series of green lights, next to a heartbeat symbol is the status light for the internet connection. Does this flash when you have problems? If so, the onboard modem has failed and that's the reason. It's a box fault and you should report it to VM - either by calling in or waiting on here for staff to respond.
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