Would incorrect size terminals cause wires to burn out?

What causes wires to burn in terminal strips is the lack of tightness between the set screw and the conductor. A loose junction will start to arc. Minor at first but as the oxidisation from the heating and cooling continues the resistance of the joint becomes higher.

As the resistance becomes higher the more amperage is needed to flow through the joint to operate the load. The heating of the conductor will get high enough to melt the insulation from the conductor. Once the insulation is gone this becomes conducive for a short circuit to occur.

If the wire gets hot enough and the circuits breaker does not trip, the wire can completely burn off from the terminal strip and the circuit will open.