The ecobee3 requires a Common (C) wire. If you had a battery powered thermostat that had the followings wires R, W, Y, G then no C-wire was needed. If you have a spare wire that can be used as the C-wire then make certain the same wire is also connected at the furnace terminal labeled "C" or "Com/24v".
If your furnace does not have a C terminal you will need to locate the wire/wires that are attached to the common side of the furnace 24V transformer.
With certain brands of furnace they will use a Molex connectors with a pigtail of wires attached. They may also be labeled to help identify the correct wires, look for something similar to this. This would be plugged into the furnace control board and the wire ends would already have your existing thermostats wires attached using wire connectors.
It also helps by referring the furnace system schematic wiring diagram, this will show where the low-voltage wires connect and where the Common(C) wire can be found.
If you don't have a spare wire to use as Common then the PEK module must be used. Refer to:
If you have only 2-wires at your thermostat, refer to this article for more information:
* Also make sure you power your HVAC system off before connecting any new wires.*
If the wiring looks correct but still not powering on, check the furnace low-voltage fuse. These can be either 2-amp, 3-amp or 5-amp.
These will be located somewhere on the furnace control module.