As far as I know your heater core won't cause smoke to come up in front of your windshield, and probably won't smell like burning plastic either. However, you're blower motor might, which is located right there. Can you hear the fan running when you turn on the blower?
If the blower is running okay you might check out the mono valve which is part of your climate control/heater system and is located near there as well. It has electrical components that I suppose could short out. It's located in front of and to the right of the battery as you face it and has heater hoses going to and from it.
There is also an electric auxiliary water pump mounted on your pass side fender well between the motor mount and coolant tank. It pumps hot water through the heater core (by way of the mono valve), but since your motor well is pretty well sealed from the cowl area by your windshield, you probably wouldn't see any smoke there if it burned out.