Plug too hot to touch after using toaster
Should the outlet be replaced?

Yes. 

I am an intensely curious person and I absolutely worship the scientific method, but even I would refrain from trying to find out exactly what the problem is here. It is a bad idea to use one's domicile infrastructure as an experimental laboratory.

The heat could be coming from a poor connection between the wires and the receptacle, or a poor contact between the receptacle slots and the plug tines. But whatever the problem, it's certainly confined to the receptacle. Just bin it.