Why don’t all objects melt and liquefy when heated sufficiently?
The problem with melting most substances is that you would be doing it in a non-innocent atmosphere and that sometimes even the functional groups inside a compound are non-innocent.

In a perfectly innocent world, everything could be molten given sufficient temperature. Given real world paper, remember that reaction $(1)$ is exothermic and the only reason why it does not spontaneously occur is the lack of activiation energy.

$$\ce{C_nH_mO_l + \frac{$2n + m - l$}{2} O2 -&gt; n CO2 + \frac{$m$}{2} H2O}\tag{1}$$

Once you supply sufficient heat, the activation energy will be overcome and the reaction suddenly becomes spontaneous; for paper, I am told that this happens at around $451~\mathrm{^\circ F}$, which is $233~\mathrm{^\circ C}$ in proper units.

Now you could suggest melting things in an inert argon atmosphere — or maybe even take helium for additional inertness. But then still you have potential reactions that can occur with organic materials. What then will happen might be called smouldering. Technically, this is no more than burning without oxygen.