Let's run the numbers:
- The wall is 700 feet high = 213 meters
- On its top, 12 knights can ride abreast and it's wider at the base. Let's say that's 25 meters wide at the top and 75 at the base, so 50 on average.
- It takes 334 kilojoules to melt 1kg of ice (if it's at 0 degrees C to begin with, but the melting energy is large compared to what it takes to heat up).
- Oak wood yields about 15 megajoules per kg when burned
- An oak tree weighs about 14 tonnes
So:
- A 100m section of wall would contain about 21350100*1000 = about 1 billion kg of ice.
- Burning 1 kg of wood can melt (at 70% efficiency) about 31 kg of ice.
- Which means you'd need to burn about 2500 fully grown trees
- In practice, efficiency would be lower, but you wouldn't need to melt the entire wall section to make it collapse. OTOH, the melting water as well as partial collapses would tend to extinguish the fire.
Given a force of many thousands of wildlings, as well as giants and Mammoths, it seems doable. But it would be a massive effort requiring many days, and giving the Night's Watch ample warning (via a gigantic smoke cloud, even if there were no patrols) and time for countermeasures.
Simply digging a tunnel would seem to be an easier and quicker alternative.