Why is the melting point of PCl3 less than that of PCl5? From just the IB chemistry bonding chapter information, you would normally deduce that $\ce{PCl3}$ and $\ce{PCl5}$ are both covalent molecules.

However, as $\ce{PCl3}$ has a dipole moment, it has dipole-dipole intermolecular attractions, implying that  $\ce{PCl3}$ has a higher melting point than $\ce{PCl5}$, which only has London dispersion forces.

So why is it that $\ce{PCl5}$ has a higher melting point than $\ce{PCl3}$ (I saw online that  $\ce{PCl5}$ is an ionic solid)?
Because $\ce{PCl5}$ does something which is not immediately obvious from its molecular formula: it autoionizes and becomes an ionic solid $\ce{PCl4+PCl6-}$. As such, it has much stronger interactions than $\ce{PCl3}$ with its mere dipole-dipole attractions, hence the higher melting point.

If not for that fact, you deduction should have worked just fine.