When should one ask a question of someone other than themselves?

Asking questions is interesting in Buddhism; the Buddha had his own version of the Socratic method, and many of the discourses are comprised of back and the Buddha asking questions in order to lead his audience to their own wisdom.

See, for example, the Anatta-lakkhana Sutta, the Buddha's second discourse:

"Bhikkhus, how do you conceive it: is form permanent or impermanent?" — "Impermanent, venerable Sir." — "Now is what is impermanent painful or pleasant?" — "Painful, venerable Sir." — "Now is what is impermanent, what is painful since subject to change, fit to be regarded thus: 'This is mine, this is I, this is my self'"? — "No, venerable sir."

(Nyanamoli, trans)

Other discourses are more traditional, wherein the Buddha is asked a question and responds in one of various ways. The Buddha himself said he responded to questions in one of four ways:

“Bhikkhus, there are these four ways of answering questions. What four? (1) There is a question to be answered categorically; (2) there is a question to be answered after making a distinction; (3) there is a question to be answered with a counter-question; and (4) there is a question to be set aside. These are the four ways of answering questions.”

-- AN 4.42 (Bodhi, trans)

Yet other teachings show the Buddha using questions out of compassion, to bring up a topic that his students are disinclined to bring to his attention out of consideration for his welfare. As the commentary explains:

jānantāpi tathāgatā pucchanti, jānantāpi na pucchanti; kālaṃ viditvā pucchanti, kālaṃ viditvā na pucchanti; atthasaṃhitaṃ tathāgatā pucchanti, no anatthasaṃhitaṃ. anatthasaṃhite setughāto tathāgatānaṃ. dvīhi ākārehi buddhā bhagavanto bhikkhū paṭipucchanti — dhammaṃ vā desessāma, sāvakānaṃ vā sikkhāpadaṃ paññapessāmāti

Sometimes, knowing, the Tathāgata asks, and sometimes, knowing, he doesn't ask. Knowing the time, he asks; knowing the time, he doesn't ask. He asks what is connected with benefit, not what is not connected with benefit; for the Tathāgata has destroyed the bridge in regards to what is not connected with benefit. With two intentions the Buddha, the Bhagava, asks questions back to Bhikkhus: "We will discourse on the dhamma, or we will establish training rules for our disciples."

-- Pārājika, verañjakaṇḍa

To this end, we can see that the Buddha himself gave some significance to asking and answering questions.

As to the purpose of questions, it is most obviously related to the way in which we acquire information, and how information informs our behaviour. Since asking questions can lead to hearing useful things, to broadening one's perspective, to reminding one of what one has forgotten, to helping one discard wrong views, and to many other good things as well, it seems obvious that questions should be asked and answered.

I can't think of a specific reference for the above paragraph, though, except to say that a read through of the Buddha's forty-five years of teaching should give a good idea of how questions are effective in helping us progress on the path. The only specific teaching on benefits of questions I can think of off the top of my head is:

“Here, student, some man or woman does not visit a recluse or a brahmin and ask: ‘Venerable sir, what is wholesome? What is unwholesome? What is blameable? What is blameless? What should be cultivated? What should not be cultivated? What kind of action will lead to my harm and suffering for a long time? What kind of action will lead to my welfare and happiness for a long time?’ Because of performing and undertaking such action…he reappears in a state of deprivation…But if instead he comes back to the human state, then wherever he is reborn he is stupid. This is the way, student, that leads to stupidity, namely, one does not visit a recluse or brahmin and ask such questions.

“But here, student, some man or woman visits a recluse or a brahmin and asks: ‘Venerable sir, what is wholesome?…What kind of action will lead to my welfare and happiness for a long time?’ Because of performing and undertaking such action…he reappears in a happy destination…But if instead he comes back to the human state, then wherever he is reborn he is wise. This is the way, student, that leads to wisdom, namely, one visits a recluse or brahmin and asks such questions.

-- MN 135 (Bodhi, trans)

So, there you have it. Asking questions leads to wisdom.

As with all sorts of learning, it is important not to mistake asking questions with actual meditation and empirical realization of the truth. On the other hand, asking questions certainly has greater benefit that simply producing theoretical knowledge; it can challenge and encourage and be quite beneficial as a means of producing wholesome mind states.