If I have a basic rule in Hubitat using a Third Reality contact sensor, or Tuya Smart Scene Remote to turn on/off lights, will that still work with no internet? Or is it only scheduled routines that will work?
If all of the devices used are 100% local, then they will continue to work when the Internet is down. Google Home, Amazon Alexa, and Apple Siri voice commands are typically processed by their respective cloud servers, and thus voice commands typically do not work when your internet connection is down.
All Zigbee and Z-Wave devices are 100% local. Many LAN connected are also 100% local (e.g. Lutron and Philips Hue.) However, some integrations do depend on the cloud to bring devices into Hubitat.
So, if you have a local button controller device, that is configured via an app (e.g. Rule Machine, Basic Rules, etc...) to control another local device, then that will work when the internet is down.
This
But best way to know for sure if you have doubts about certain devices, especially WiFi devices, just unplug / turn off your modem and test your devices to see if they keep working or not.