Will the Lyric T5 Wi-Fi Thermostat work without a C wire being installed?

No, the Honeywell Lyric T5 Wi-Fi Thermostat requires a C wire or Common wire be connected to the C terminal in the thermostat and at the equipment terminal.

The C wire or Common wire is needed to provide 24-volt power to your Honeywell T5 Wi-Fi or connected thermostat.

There is a temporary solution and a couple of permanent options that you can implement if you do not have a C wire, as well as contacting an HVAC professional near you for additional help with your Honeywell T5 Wi-Fi Thermostat install.