Diagnosis
This is almost always a wiring issue at your furnace's control board, not the ecobee itself. The ecobee needs a constant 24VAC from the R and C (common) wires, and your old thermostat likely didn't use a C wire.
How to Fix It
Go to your furnace. Open the panel and locate the control board. You need to find the terminal labeled 'C' (common). There should be a bundle of unused wires tucked back there. Take the blue wire from that bundle (or whatever color you ran to the C terminal at the thermostat), strip the end, and secure it under the 'C' screw on the board. Turn the furnace power back on, then the ecobee should boot.
If That Didn't Work
If you can't find a spare wire, you need a 'PEK' (Power Extender Kit). It comes in the box with the ecobee Lite. It lets you power the thermostat without a dedicated C wire by re-purposing the fan wire at the furnace board. The installation guide for the PEK is very specific—follow it exactly.
Heads Up
When you shut off power to the furnace, flip the switch on the unit itself AND the breaker. It's easy to miss the little furnace switch that looks like a light switch.