There is, of course, always the possibility of the cable failing when you are not in attendance.
Electric irons are unusual in terms of household appliances because they have a high current demand combined with a need for flexibility due to the way they are used.
Other handheld high current demand items tend not to be used for long periods of time in the way an iron is.
The power cord is usually a number of fine stranded cores, each bundle of cores is wrapped in a soft silicone rubber casing, which is not as mechanically strong as pvc insulation.
This type of insulation is required for heat resistance and flexibility.
The outer casing is usually a braided mesh tube, which is also very flexible and strong.
When an iron is in use, the outer braid over time will very often develop openings near the handle of the iron, and the inner cores force their way out, bulging at this point.
It is not unusual for the inner cores to slowly wear away and for the actual conductors to be bared, possibly shorting out against each other- good because it trips the circuit- bad if it just causes a partial short, because then heat will be developed around the cable entry and a fire can result.
The other reason why it is not a good idea for an electric iron to be left on is the action of the thermostat.
Temperature of an iron is nearly always controlled by a bi-metallic heat sensitve switch. As the two dissimlilar metals expand with heat, one will expand more than another, causing the strip to bend away from the more expansive material, this metallic strip is used to move a contact which makes and breaks the main circuit.
Unfortunately, over several years, the constant on/off action of the thermostat will erode the contact, whose surface will become pitted, and eventually it will weld itself onto its opposite countepart.
Once that contact has welded the iron is permanently on, and it will continue to get hotter and hotter.
Manufacturers do fit a high temperature safety cut out, but this does not permanently break the circuit, as soon as the iron turns off using the safety cut out, it will cool a little, and the iron will be turned on again.
The safety cut out is not desgned to operate this way, and is much more likely to weld closed, when this happens that is absolutely nothing to limit the rise in the temperature of the iron, sooner or later something will melt, and a fire is a high probabilty.
If you are in attendance, you would very soon notice something was amiss if the iron was working only on the safety cut out, it would be very much hotter than normal, it would damage any clothes it came into contact with, it would have that characteristic electrical overheat smell, in short, you would soon turn it off.
If you are not in attendance you of course cannot do this.
Leaving an iron on when not in attendance is an extremely bad idea.
In the UK our electrical system does make this less likely, our plugs are physically large compared to the US counterpart.This is because our plugs contain a fuse that is suitable for the particular appliance it supplies, whereas in the US the fuse is in the supply box and is rated for all the appliances likely to be connected to that circuit.
The result is that the UK device will have something like a 10Amp or 13Amp fuse, whereas the US will be protected by a far larger value fuse.
The UK device will be more likely to trip the fuse in its power cord plug than the US device will trip the fuse in the supply board.