Question: Is it permissible to ask the same question to more than one Rabbi?
Answer: It depends on what you are asking. The Gemara in Avodah Zarah (7a) says that one who asks a Rabbi a question and he (the Rabbi) declares it impure may not ask another Rabbi who will declare it pure, and one who asks a Rabbi a question and he declares it forbidden may not ask another Rabbi who will declare it permissible. This ruling is quoted in the Shulchan Aruch (Yoreh Deah 242:31). Why is it forbidden to ask the same question a second time to a different Rabbi? Some explain that it is because of the honor of the first Rabbi (Rashi to Niddah 20b): You asked a question and don't like the answer so you go to a different Rabbi?! You are shaming the first Rabbi! Others explain that when the first Rabbi rules, the object on which he ruled now has the status which he placed upon it. This means that if I ask a Rabbi if something is kosher or not and he rules that it is not Kosher, the ruling of another Rabbi cannot change it. The Halachah follows the second explanation (This is the opinion of most Rishonim, including Ra'avad, Ramban, Rashba quoted in the Ran Avodah Zarah ibid. and Rosh, ibid. 1:3). Therefore, when I ask a Rabbi a question about a piece of meat, the meat has the status of his ruling, but if I have another piece of meat and I have the same question, I can ask a different Rabbi.
There are also questions regarding a person's activities: How should I act in a given situation? A Rabbi's ruling fixes the status of an object, but not the status of a person's activities. Regarding an object, you can only ask one Rabbi, but regarding a person's conduct, you can ask various Rabbis. Even in the case of an object, if I fervently want to ask a second Rabbi, I can, as long as I tell him that I already asked the first Rabbi. If the second Rabbi so desires, he can talk to the first Rabbi and try to convince him to change his mind (Rama ibid.).
I remember that someone once asked me a question regarding the laws of Family Purity and I answered: she is impure. The questioner went and asked Ha-Rav Mordechai Eliyahu. Ha-Rav Eliyahu called me and said: "Rav, look at it from this perspective and that perspective." I then understood that it was permissible to be lenient and I said: "I retract, she is pure." Furthermore, it is obvious that someone who asks a theoretical question may ask as many Rabbis as he wants. You may also ask questions to different Rabbis at different times, since all Rabbis are Torah.
By the way, if someone accidently asked the wrong Rabbi a question, it is permissible to re-ask the question. If he intended to ask a Rabbi in general (and not a specific Rabbi), he must follow his answer. And it once happened that a couple had a question on Shabbat night about the laws of Family Purity. Since they lived near to Ha-Rav Ovadiah Yosef, the wife went to his apartment building, but accidentally went to the floor above Rav Ovadiah, where Ha-Rav Ben Tzion Abba Shaul lived, and he ruled that it was forbidden. When she returned home, the husband understood that his wife had made a mistake. He went on his own to Rav Ovadiah, who permitted it, and related that his wife had accidentally asked Rav Ben Tzion Abba who prohibited it. Rav Ovadiah said: Rebbe Ben Tzion is a Gaon in Halachah, but my opinion in this case is that it is permissible. Therefore, if you originally intended to ask me, it is permissible, but if you intended to ask any Rabbi, it is forbidden, especially since you asked Chacham Ben Tzion, and I cannot permit what he did not (Maran by Ha-Rav Michal Shtern pp. 247-248).