A motor can give you a brief spike that's outside of rating. If you ever turned on an appliance with powerful motor and noticed lights dimming, that's that. How much power the motor will draw when running at full speed will depend on the amount of mechanical work the motor is doing. If it's spinning freely, it will only draw enough power to overcome friction and some to be wasted on resistance of the coil.
A heater is trickier. Like I said, the resistance will increase as the heating element heats up. That means power consumption will be higher when it's cold, but not by much. The way it will maintain temperature, however, is by turning heating element on and off.
In terms of what will actually limit the current, it's going to be very different depending on what the device does and whether we are talking about AC only or if we consider what happens in DC stages of devices that have them. Seriously, this isn't something that can be just covered in a few lines of text. If you are interested, start learning more about electrical circuits.