Adi Shankaracharya says that low-caste people are not eligible for Jnana. Here is what he says in this section of his Brahma Sutra Bhashya:
(In the preceding adhikarana) the exclusiveness of the claim of men to knowledge has been refuted, and it has been declared that the gods, &c. also possess such a claim. The present adhikarana is entered on for the purpose of removing the doubt whether, as the exclusiveness of the claim of twice-born men is capable of refutation, the Sûdras also possess such a claim.... To this we reply that the Sûdras have no such claim, on account of their not studying the Veda. A person who has studied the Veda and understood its sense is indeed qualified for Vedic matters; but a Sûdra does not study the Veda, for such study demands as its antecedent the upanayana-ceremony, and that ceremony belongs to the three (higher) castes only. The mere circumstance of being in a condition of desire does not furnish a reason for qualification, if capability is absent. Mere temporal capability again does not constitute a reason for qualification, spiritual capability being required in spiritual matters. And spiritual capability is (in the case of the Sûdras) excluded by their being excluded from the study of the Veda.--The Vedic statement, moreover, that the Sûdra is unfit for sacrifices intimates, because founded on reasoning, that he is unfit for knowledge also; for the argumentation is the same in both cases.
Here is what he says in another section of the Brahma Sutra Bhashya:
The Sûdras are not qualified for that reason also that Smriti prohibits their hearing the Veda, their studying the Veda, and their understanding and performing Vedic matters. The prohibition of hearing the Veda is conveyed by the following passages: 'The ears of him who hears the Veda are to be filled with (molten) lead and lac,' and 'For a Sûdra is (like) a cemetery, therefore (the Veda) is not to be read in the vicinity of a Sûdra.' From this latter passage the prohibition of studying the Veda results at once; for how should he study Scripture in whose vicinity it is not even to be read? There is, moreover, an express prohibition (of the Sûdras studying the Veda). 'His tongue is to be slit if he pronounces it; his body is to be cut through if he preserves it.' The prohibitions of hearing and studying the Veda already imply the prohibition of the knowledge and performance of Vedic matters; there are, however, express prohibitions also, such as 'he is not to impart knowledge to the Sûdra,' and 'to the twice-born belong study, sacrifice, and the bestowal of gifts.'
But he does give one exception to this, namely that in rare cases a low-caste person can spontaneously acquire Jnana by virtue of their past births:
From those Sûdras, however, who, like Vidura and 'the religious hunter,' acquire knowledge in consequence of the after effects of former deeds, the fruit of their knowledge cannot be withheld, since knowledge in all cases brings about its fruit.
Adi Shankaracharya also says that Brahmavadini women (see my answer here) like Yagnavalkya's wife Gargi are eligible for Jnana:
Even a person who because he does not belong to an âsrama stands between, as it were, is qualified for knowledge. 'For that is seen.' For we meet with scriptural passages declaring that persons of that class - such as Raikva and the daughter of Vachaknu - possessed the knowledge of Brahman.
But notwithstanding all of that, in this section of his Bhagavad Gita Bhashya he says that those who perform Sharanagati, even low-caste people and women, will attain Moksha:
Hi, for; O son of Prtha, ye api, even those; papayonayah syuh, who are born of sin;-as to who they are, the Lord says-striyah, women; vaisyah, Vaisyas, tatha, as also; sudrah, Sudras; te api, even they; yanti, reach, go to; the param, highest; gatim, Goal vyapasritya, by taking shelter; mam, under Me-by accepting Me as their refuge.
He doesn't elaborate on this, though; he just repeats what the Gita verse is saying.