Can I safely reconnect thermostat wire on furnace controller?
The W terminal for a thermostat connection (normally also mated with a white wire) is the one used to initiate a "heat call" from the thermostat. If that wire was left open then it makes sense that the "cool call" and "fan" functions would have worked just fine during the summer months but that now heating does not.

It is possible that a furnace checkout person had opened the white wire so that they could manually trigger a heat call with a jumper lead during the testing and checkout of the furnace. This would have been done to prevent any surprises from the thermostat when service was in process. 

My take is that it was an oversight to not have reconnected the white wire as the service call was wrapping up. You should have no problems connecting the wire back up when the furnace power is Off.