Generally, yes. If the wires get to hot, they will melt, spark together, and make a fireworks show just before catching fire. The main cause of this problem is the manufacturer using wire that is not the correct gauge for the power required. This unfortunately is not a fixable situation for most people. If the unit is relatively new, i would suggest contacting the manufacturer about your concerns. If it is older, please replace it with a newer model.
There is more than one possible cause. One of which you can fix. If the plug prongs or the contacts in the receptacle get oxidized, usually caused by some burning or arcing or just plain aging, this will produce a point of electrical resistance. At this point power is consumed so heat results. It can be enough to melt the plug and wires!
The other possibility is that, at the point where the wire meets the plug, several of the internal strands of conductor wire are broken. Again, this causes a resistance point where heat is produced.
In the first case, you can just clean up the contacts. In the second case you must remove and replace the plug. Cut the wire back to a fresh point.
Another possibility is that the electrical outlet is old and worn. If the toaster is plugged into an outlet where the plug prongs don't fit tightly, then heat from arcing can occur right where the prongs meet the contacts in the outlet. This is more likely to happen with appliances which draw heavier current, such as the toaster or a hair dryer. If the outlet does seem old and loose then try another one and see if the plug prongs and cord stay cooler. Some warmth of the plug and cord is normal, but it shouldn't feel hot.
If necessary, you can replace the outlet for minimal cost, if you understand how to do so (be very careful or consult an electrician if you're not sure).
If the outlet is newer and still tight, and there's no visible damage as Damian suggested, then it could mean that Tracy's answer is correct, that the manufacturer used a smaller gauge cord than ideal.