Most heaters on high are 1500W. That's close to 13A depending on your line voltage. They are not designed for continuous use. This power will make the line cord (HPN typ.) get warm to the touch. The blades on the power plug will be very warm to touch as well under long runs with this power draw. Also be aware that most receptacles employ series/parallel wiring so if you're plugging in to a receptacle at "the end" of the line, many other sockets - even if not in use - can be a cause of heating and reduced line voltage. A simple check with a forward looking infrared radiometer (FLIR) will reveal hot receptacles. Back wired ones are frequent offenders. Also, improperly terminated aluminum wiring will cause heating.
Heater coils on old units (bare nichrome ribbons or coils) on high settings could glow cherry red. Usually fan forced heaters with reduced airflow will have coils glowing brighter. They will burn out fairly quickly if left running like this. The mica insulators will also degrade faster and if the coil touches the chassis and shorts it will glow white hot and spark all nice and hopefully no one has put a penny under the fuse!
Ceramic heaters do not glow at all and should be kept free of dust and other debris. These types like the original Pelonis disc furnace had foam filters that needed to be cleaned.
They also had very high inrush current and could pull 25A or more for a few seconds when cold started. This would blow non slow blow fuses if the circuit was protected by them which was a pita.
Actually, plug in heaters are rated for continuous use, at least on the premise wiring branch circuit side. That is typically why they top out at 1500w. The NEC requires a branch breaker be sized 125% of the load for a continuous load. This is fudged a bit, but for a 1500w heater you would adjust the load to about 15.6 amps at 120v. Which is just a hair over the standard 15a branch circuits many homes still have for general purpose receptacles. This is also why home hair dryers can be as high as 2000 watts, as 20 amp branch circuits have been required in bathrooms for quite a while.
The problem lies in a cheaper manufacturer with either poor quality control of the cord/plug assembly, or just barely sizing the wire to meet voltage drop requirements. Any extra resistance in the branch wiring through splices or terminations exacerbate this problem.
I have used 2 heaters with heavy duty 3 prong plugs at 1500w for over a week straight and not had any plug warming, one was even on a 50' 12AWG extension cord. This was at a new construction site with 20amp branch breakers, and low ambient temps.
The topic of "backstabbed" receptacles is something debated on the electrical forum I frequent, but it is generally agreed that the builder grade receptacles with tiny fingers on the push-in rear connections should not be used by a professional electrician.
Everything else is good advice and spot on, although if people are still using a fuse box in a residential setting, then a plug-in heater is the least of their worries. Modern circuit breakers of course can handle inrush currents, as they have both a magnetic and thermal cutout.