When subjected to heat, why do some objects melt while other objects burn?

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To answer this question fully let's explain why things melt and why things burn. So let's start with melting.

Melting is what is called a phase transition. That is when a substance changes from solid to liquid to gas. Temperature affects how much the atoms of a substance move, if they are really hot they will be too energetic to hold together at all and be a gas. At lower temperatures, they move less so they can hold together but do not have any solid form, a liquid. At even lower temperatures, they form crystals, solid forms, and become a solid. These changes are caused temperature and pressure. Floating in space, an object is under no pressure, space is a vacuum. However on earth. We on the surface have about 15 pounds of atmosphere pushing down on every square inch of space. Every object's phase transition can be described in a chart like this[1]  . That shows how temperature and pressure affect phase transition, notice how when air pressure is lower, water also boils below 212F/100C.

Burning is a totally different physical property, that describes combustion, and (at least to me) is a little more complicated to explain. This is a property separate from phase transition that describes chemical reactions called combustion. These reactions give off energy in the form of heat or light, and require an oxidizing agent What on earth is that? The most familiar oxidizing agent is oxygen, it makes stuff burn. "Yeah, genius," you say, "but how?" Now we have to delve a little more into chemistry. Most atoms are held together by covalent bonds, atoms share one or more electrons, becoming a new substance, like water plus hydrogen makes water, the hydrogen and oxygen share electrons which bind them into water.

Burning is the result of a chemical reaction where the oxidizing agent accepts electrons, creating new compounds that are smaller than the ones you started with. So when you burn sugar carbon dioxide and water are created. (your body also does the same thing at much lower temperatures, burning sugar to make water and CO2. The heat provides enough physical energy to allow the reaction to occur.

This is how we send rockets into space btw. The rockets have tanks of compressed hydrogen and oxygen, whch have nozzles which meet at the end of the rocket. When you're ready for liftoff, you open the tanks, add a spark, and BOOM. That big trail of gas that comes out of the end? That's water vapor.

So when you strike a match, the friction raises the temperature of the phosphorus in the match head to create an oxidizing reaction, making a flame, since combustion creates heat, this starts to heat the rest of the match, making it burn.

To answer your question water can burn, if you add pure phosphorus, it will combust and make an explosive chemical reaction. Wood can liquify, heat it and it'll burn first, turn to charcoal, heat that enough, and it will liquify. It happens in the earth, plants and animals dead for millions of years, get buried are exposed to tremendous heat and pressure and become raw petroleum.