Question: Does a person who learns Torah but does not teach, answer questions, serve as a Rabbi, etc. have influence on a community?
Answer: Certainly. The Gemara at the end of Ketubot (104a) relates that Rebbe Yehudah Ha-Nasi was suffering terribly. His maidservant saw and prayed that he should die but the Sages prayed that he should not die. In the book "Midbar Shur," in his eulogy for Ha-Rav Yitzchak Elchanan Spector (pp. 332-336), Maran Ha-Rav Kook asks: Why did the Sages pray that he should not die? Their view is difficult to understand. After all, Rebbe Yehudah Ha-Nasi was bed-ridden, suffering, could not teach or give halachic rulings, and was seemingly of no benefit to this world. If he would ascend on High, he would continue to teach Torah there. So why didn't they pray for him to die? Maran Ha-Rav Kook explains that the influence of a great Torah scholar is not only through his teaching, halachic rulings, etc., but also in the presence of his holy soul in this world. The fact that his soul is located in this world brings blessing, even when he is unable to provide practical benefit, is closed in a room and cannot converse with others. This is similar to the Vilna Gaon, who for many years was closed in a room learning Torah. The world without Rebbe Yehudah Ha-Nasi is not the same as a world with Rebbe Yehudah Ha-Nasi.
And when Rabbenu Ha-Rav Tzvi Yehudah taught us this idea, he would say that Maran Ha-Rav Kook also suffered greatly, and he told him: Each and every moment that Abba is in this world, despite the suffering, he brings it light. And Rabbenu would relate this with tears in his eyes.
This insight also answers a difficulty in Rashi (see Klei Yakar). At the beginning of Parashat Vayetze, the Torah says: "And Yaakov left Beer Sheva and went to Charan" (Bereshit 28:10). Rashi makes the comment that when a Tzadik leaves a city it makes an impression. The commentators ask: Why does Rashi comment on Yaakov leaving the city? What about when Avraham left to go to Egypt? What about when Yitzchak went to Gerar? They were certainly Tzadikim! Why doesn't Rashi point out that their departure made an impression? The answer is that it is obvious that Avraham's departure made an impression. He was a man of Chesed and welcomed everyone in! His tent had openings on all sides. And when Yitzchak left, it certainly made an impression. He was well-known and planted all over Eretz Yisrael. But Yaakov, at this point in his life, was always learning Torah in the Beit Midrash, "A simple man who learned in tents" (Bereshit 25:27). He was closed off from everyone. We might think that if someone who "just" learns Torah and doesn't interact with others leaves a city, it won't make an impression. Rashi therefore mentions this idea in connection to Yaakov, as opposed to Avraham and Yitzchak, to teach us that the opposite is true: Someone who learns Torah has an incredible influence on his place even if he is simply learning Torah on his own.