In general, HDHomeRun devices "phone home" every 6 hours or so to update their
DeviceAuth token, which SiliconDust uses to authorize for guide access, as well as to allow your devices to register with their
my.hdhomerun.com website. In practice, you can use a HDHomeRun tuner without it being connected to the internet—even the Prime—which I successfully did for years.
That particular model is different. The HDHR3 and earlier models, except for the Prime, are considered legacy devices. They do not store their channel lineup on the device. Instead, a channel scan must be made from a Windows computer with SiliconDust's software, which then uploads the lineup to SD's servers. Once that initial scan is done, then if the HDHR3 model has internet access, it will emulate a newer tuner that can maintain its own lineup. This is not SD disabling a feature, as these devices never maintained their own channel lists; instead it is SD offering a service for older devices to function as the newer models. (SiliconDust's own software needs this new emulation mode to work with these older devices, as they longer longer support legacy operation in their software.)
However, not all software needs this lineup to maintain a channel list. Tvheadend and MythTV both perform their own channel scans and store the lineups separate from the one provided by the device. I am uncertain if Channels can maintain its own scan/lineup for these legacy devices, but their operation in general is not hindered by not being connected to the internet.