If your ecobee was previously powered on and working fine, a sudden black screen usually indicates insufficient voltage going to the thermostat. The ecobee draws power from the furnace, so any disruption to this power source will result in a loss of power at the thermostat. If you have a multimeter, you can test the voltage between RC and C at the ecobee—anything less than 24 VAC is insufficient.
Check for the following:
A tripped circuit: Check your breaker box to ensure that the circuit powering the furnace isn’t tripped. If it is, turn it back on to restore power to the furnace and the ecobee should power back on.
A blown fuse on the furnace control board: Open the panel on your furnace to expose the control panel where your thermostat wires are connected. Most systems have a purple 3 or an orange 5 amp fuse on the board—check this fuse to ensure that it’s intact. A blown fuse will have to be replaced. Once a blown fuse is replaced, the ecobee should power back on.
A tripped high-limit temperature sensor: This is usually accompanied by rebooting at the thermostat during a call for heat which may also result in a complete loss of power to the thermostat. To learn more about what causes a high-limit temperature sensor visit: My furnace won’t turn on! How can I troubleshoot?
A tripped float switch: This is usually accompanied by rebooting during a call for cooling which may result in a loss of power completely. Inspect your float switch/condensation pump/drip pan (located at the furnace) to see if there's an excess of water or any blockage preventing the water from draining. Once this is addressed, the ecobee should power back on.
If you’re installing your ecobee for the first time and it’s not powering on, please refer to the following article: My ecobee is not powering up after installation. What could be the reason?