(a) I heard from a Rav years ago that it says somewhere in the Gemara or other sources that if you look in a cat's eyes, it will steal your knowledge of Torah. I have never found the source for this, or any cat-related superstitions. Are there any sources, even though I realize the question is somewhat ridiculous?
(b) Also, is this a case where a superstition (like that a black cat crossing your path is bad luck) is avodah zorah and should be ignored?
(a) Hagaon Rav Chaim Kanievsky has a wonderful pamphlet called "Sefer Zikaron" (printed with his Si'ach ha'Sadeh) in which he lists every time Chazal mention that a particular practice will either cause one to forget or to remember what he learned. He does not list looking into a cat's eyes, so we may safely assume that there is no source for such a statement in Chazal.
(b) Since it does not appear in Chazal nor has it been shown to be true, it should certainly be ignored. If something does appear in Chazal, it obviously should not be ignored. I will include below something that we wrote on this subject in the past.
Insights to Chulin 77b
4) "DARCHEI HA'EMORI" IN AN ACT DONE FOR "REFU'AH"
OPINIONS: Abaye and Rava rule that any act that is done for the purpose of Refu'ah does not constitute Darchei ha'Emori, while any act that is not done for the purpose of Refu'ah constitutes Darchei ha'Emori. What is considered an act done for Refu'ah?
(a) RASHI (DH Yesh Bo) says that an act done for the purpose of Refu'ah includes using any liquid or potion (that has no quantifiable medicinal value) for healing, or chanting an incantation over a wound. Rashi (DH Ein Bo) explains that an act that is not done for the purpose of Refu'ah refers to any act that is not done "on a sick area," such as burying a Shilya at a junction in the road. The PANIM ME'IROS (1:36) understands that Rashi means to say that we are allowed to do an act, even for the sake of healing, directly over the body of a sick person, but we may not do an act from a distance. For this reason, the Panim Me'iros rules that it is forbidden to make a amulet to be hung in a tree to help a baby sleep better. (See, however, the PISCHEI TESHUVAH (YD 179:5), who disagrees with the understanding of the Panim Me'iros (his grandfather) of the words of Rashi.)
(b) RASHI in Shabbos (67a, DH she'Yesh) defines Refu'ah as an act that is an effective treatment for the illness from an empirical perspective. This seems to exclude an incantation that is whispered over a wound in order to heal it.
The RAMBAM (Moreh Nevuchim) also writes that the act must have some medicinal quality to its healing ability in order to be permitted. The RASHBA (Teshuvos 1:413) questions the Rambam's opinion from the case in the Mishnah in Shabbos (67a). The Mishnah quotes Rebbi Meir who permits one to wear a fox tooth in order to help him sleep. The Rambam himself rules in accordance with the opinion of Rebbi Meir (Hilchos Shabbos 19:13), which is difficult, since there seems to be no empirical therapeutic effectiveness in wearing a fox tooth.
(c) The RAN here quotes Rashi in Shabbos and asks (like the Rashba) that the Gemara permits one to wear a fox tooth to help him sleep, even though its therapeutic property is not understood from a scientific perspective. The Ran therefore defines an act done for the sake of Refu'ah as any act that we know works to heal, even if it works metaphysically. An act that is not done for the sake of Refu'ah is an act that has no known results.
I wonder if this has any connection with the Gemara in Horiyos 13a and 13b with regard to a cat being forgetful and that one who eats from what a cat ate could forget what he learned, for a full discussion on this see Sefer Shmiras Haguf vHanefesh chapter 16.
Other cat related issues discussed in that Sefer are:
Walking barefoot in a house where there is a cat, ibid footnote 9.
Not feeding blood to a cat, chapter 35:1
Not killing a black cat, chapter 249:3
Not threatning a child that a cat (or other animals) will take you, chapter 229