The question as you posed it was whether "lip balm" in general can make skin dryer. Lip balm can have any number of ingredients. A long list can be found here. ChapStick, the most common brand of lip balm, contains petroleum jelly/petrolatum. A textbook on the subject, Dry Skin and Moisturizers, states: "Petrolatum blocks the evaporation of water from the skin (transepidermal water loss, TEWL), thus keeping the stratum corneum well hydrated." (In other words, ChapStick is mostly the same as Vaseline, and there is nothing harmful about putting Vaseline on your lips.)
A scientific study cited on that page showed petrolatum to have superior abilities at retaining moisture in the skin compared to herbal remedies like the active ingredients (?) of Burt's Bees organic lip balm. Burt's Bees provides an example active ingredient of salicylic acid. Of this, the Dry Skin and Moisturizers book states: "It should be remembered that salicylic acid preparations may be irritating and can cause systemic toxicity if applied to large areas of damaged skin." Elsewhere in the book, it's recommended to avoid giving salicylic acid to children. And there's more:
The treatment of the various familial ichthyoses requires exfoliating
agents, often wrongly called keratolytics, aimed at removing excessive
horn. These include α-hydroxy acids, salicylic acid, propylene
glycol, urea, and mixtures of these. These same agents may be
harmful on the dry face, often causing neurosensory reactions such as
itching, burning, and stinging as well as scaling and redness.
The two doctors quoted in this question and other answer may also be discussing the many possible adverse effects associated with herbal balms. Some flavors of ChapStick contain menthol or camphor, which can cause allergic or other reactions in some people. A highly cited summary paper on this is available: Adverse effects of herbal drugs in dermatology (PDF)
Other popular remedies that can cause dermatological side-effects include St John’s Wort, kava, aloe vera, eucalyptus, camphor, henna and yohimbine.