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I learned that, a phenol will not react (or react very slowly) with $\ce{PCl5}$ due to its stabilized structure.
How about the $\ce{-OH}$ group on the benzene ring of 4-hydroxybenzoic acid? Will that $\ce{-OH}$ group react with $\ce{PCl5}$? If the answer is yes, is it anything to do with the $\ce{-COOH}$ group?
a phenol will not react (or react very slowly) with $\ce{PCl5}$
That's not true. Phenol and $\ce{PCl5}$ do react, producing mixture of compounds of formula $\ce{PCl_n(C6H5O)_{5-n}}$ (with $n > 1$ ). It is even possible to produce $\ce{(C6H5O)5P}$![1]
However, unlike in alcohols, in phenols:
the carbon–oxygen bond is much stronger, having a bond order of more than one and being formed with $\mathrm{sp^2}$ hybridized orbitals
the formation of the corresponding transition state for nucleophilic substitution would destroy the aromaticity of the benzene ring.
Thus, the formation of chlorobenzene does not proceed at a significant rate.
It is still possible to replace the phenol group by a chloride atom in some cases. For example, it is possible to prepare picrylchloride (1,3,5-trinitrochlorobenzene) by reaction of pyridine picrate and $\ce{PCl5}$, since three nitro groups greatly increase the reactivity of the phenyl fragment in nucleophilic aromatic substitution reactions.[2] However, as a general rule, benzene derivatives are not active in nucleophilic substitution except if there are a lot of strong π-electron withdrawing groups in the ring.
As for 4-hydroxybenzoic acid, my guess is that it should react as both phenol (as above) and benzoic acid (which forms benzoyl chloride).
While I was unable to find precise information on para-hydroxybenzoic acid, there is some info on ortho-hydroxybenzoic (salicylic acid). $\ce{PCl5}$ replaces the $\ce{OH}$ fragment of the carboxylic acid group with chlorine. The phosphorus oxochloride then reacts with the phenolic group, producing an $\ce{O-POCl2}$ fragment. The information, however, does not look absolutely trustworthy. Still, assuming it is correct and extrapolating it onto para-hydroxybenzoic acid, I'd say that the product should be $\ce{Cl-C(O)-C6H4-O-P(O)Cl2}$.
[1]: Ramirez, F.; Bigler, A. J.; Smith, C. P. Pentaphenoxyphosphorane. J. Am. Chem. Soc. 1968, 90 (13), 3507–3511. DOI: 10.1021/ja01015a038.
[2]: Boyer, R.; Spencer, E. Y.; Wright, G. F. Can. J. Chem. 1946, 24b (5), 200–203. DOI: 10.1139/cjr46b-025.
