In the name of God, the Most Gracious, the Most Merciful.

Has Islam prohibited questioning?

Allah, the Exalted, said in Surah Al-Ma'idah: "O you who have believed, do not ask about things which, if they are shown to you, will distress you. But if you ask about them while the Qur’ān is being revealed, they will be shown to you. Allāh has pardoned it [i.e., that which is past]; and Allāh is Forgiving and Forbearing." (Al-Ma'idah 101).

This verse has prohibited the believers from asking the Prophet, peace be upon him, about matters that God has remained silent about in His Book and has overlooked; for probing and dissecting such questions might lead to their prohibition, which would burden them. It has also warned them against inquiring about things that are hidden from them, even if they seem troubling, such as questioning the legitimacy of their lineage to their fathers.

And it happened. Allah's Messenger (ﷺ) came out as the sun declined at midday and offered the Zuhr prayer. He then stood on the pulpit and spoke about the Hour (Day of Judgment) and said that in it there would be tremendous things. He then said, "Whoever likes to ask me about anything he can do so and I shall reply as long as I am at this place of mine. Most of the people wept and the Prophet (ﷺ) said repeatedly, "Ask me." `Abdullah bin Hudhafa As-Sahmi stood up and said, "Who is my father?" The Prophet (ﷺ) said, "Your father is Hudhafa." The Prophet (ﷺ) repeatedly said, "Ask me." Then `Umar knelt before him and said, "We are pleased with Allah as our Lord, Islam as our religion, and Muhammad as our Prophet." The Prophet then became quiet and said, "Paradise and Hell-fire were displayed in front of me on this wall just now and I have never seen a better thing (than the former) and a worse thing (than the latter). [Sahih al-Bukhari 540]

Imam Muslim, may Allah have mercy on him, said after narrating this hadith: "Ibn Shihab reported to me that Ubaidullah bin Abdullah bin Utbah said: The mother of Abdullah bin Hudhafah said to Abdullah bin Hudhafah: 'I have never heard of a son more disobedient than you.'" Do you feel secure that your mother has not committed some of the things that the women of the pre-Islamic era used to do, and that she would be exposed before the eyes of the people? Abdullah ibn Hudhafa said: "By God, if a black slave were to catch up with me, I would catch up with him."
It is concluded from "Sahih Muslim" (4 / 1832).
 

Ash-Shawkaani (may Allah have mercy on him) said in his commentary on the verse in al-Maa’idah – “O you who believe! Ask not about things which, if made plain to you, may cause you trouble” –: That means: do not ask about things that you do not need to ask about and that are none of your concern in matters of your religion. The words “if made plain to you, may cause you trouble” are describing those things. In other words: do not ask about things of this nature, which if they were made plain to you and were made obligatory upon you, would cause you trouble. Allah forbade them to ask the Messenger of Allah (blessings and peace of Allah be upon him) a great deal, because asking about matters that do not concern one or that one does not need to know may be a cause of it being made obligatory upon the one who asks the question and others. The words “But if you ask about them while the Qur’an is being revealed, they will be made plain to you” also describe those things. What is meant is: do not ask about things, for if you ask about them whilst the Qur’an is being revealed, which is whilst the Messenger of Allah (blessings and peace of Allah be upon him) is still among you and the Revelation is still coming down to him, they will be made plain to you, i.e., they will be explained to you  by means of the answer that the Prophet (blessings and peace of Allah be upon him) gives you or what is enjoined upon you by the Revelation. Thus that will be a cause of hardship and of making obligatory something that was not obligatory, or of the prohibition of that which was not prohibited. This is unlike asking about things after the Revelation ceased with the death of the Messenger of Allah (blessings and peace of Allah be upon him), for then no obligations or prohibitions would come about because of asking. End quote from Fath al-Qadeer by ash-Shawkaani (2/92). 

A number of things are known from this: 

Firstly:

The prohibition mentioned in the verse applies specifically to the time of the Prophet (blessings and peace of Allah be upon him), which was the time when Revelation was still coming down, unlike times that came after that. Hence this does not mean that we should not seek Islamic knowledge and learn that which will benefit us or that we need to know with regard to matters of this life and the hereafter. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

it is truly unfortunate that some people say that you should not ask lest you be told of something that will cause you hardship, then they misinterpret the verse in which Allah, may He be exalted, says (interpretation of the meaning): “O you who believe! Ask not about things which, if made plain to you, may cause you trouble” [al-Maa’idah 5:101]. Rather the prohibition on asking was applicable only at the time when the Revelation was coming down, at which time it was possible that the rulings could be renewed or changed. But after the death of the Messenger of Allah (blessings and peace of Allah be upon him and his family), it is obligatory for the individual to ask about everything that he needs to know about matters of his religion. End quote from Fataawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen (24/223) 

Secondly:

The things that they were forbidden to ask about were: 

1. Things that had no bearing on their actions and were of no benefit in religious terms.

2. Things that Islam said nothing about, out of mercy to people, not out of forgetfulness, because asking too many questions about that could lead to them being forbidden, which would cause hardship for people. Al-Bukhaari (6745) and Muslim (4349) narrated that Sa‘d ibn Abi Waqqaas (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “One of the worst sins a Muslim could commit is asking about something that was not prohibited to the Muslims, but it became prohibited to them because of his asking.”

3. Things that were unknown to those who asked about them, which if they were made plain to them would cause them trouble, as we have seen above in the hadeeth of ‘Abdullah ibn Hudhaafah.

Thirdly:

With regard to matters that are connected to the explanation of a shar‘i ruling, Allah has instructed us to ask about them, as He says (interpretation of the meaning): “So ask those who possess knowledge if you do not know” [an-Nahl 16:43]. 

Shaykh as-Sa‘di (may Allah have mercy on him) said in his commentary on the verse in al-Maa’idah:

Allah forbids His believing slaves to ask about things which, if they are made plain to them, will cause them trouble and grief, such as when some of the Muslims asked the Messenger of Allah (blessings and peace of Allah be upon him) about their fathers, or about their situation in Paradise or Hell. Perhaps if these matters were made plain to the one who asked, there would be nothing good for him in it. Another example is when they asked about things that had not happened, or they asked about things which would lead to stricter rulings in sharee‘ah that could cause hardship for the ummah, or they asked about matters that do not concern them. Asking about these and similar matters is what is forbidden. As for asking questions that will not result in any such thing, this is what is enjoined, as Allah, may He be exalted, says (interpretation of the meaning): “So ask those who possess knowledge if you do not know” [an-Nahl 16:43]. 

And about the hadith: The Prophet (ﷺ) said, "Allah has forbidden you: disobedience to your mothers, to withhold (what you should give), or demand (what you do not deserve), and to bury your daughters alive. And Allah dislikes idle talk, to ask too many questions (for things which will be of no benefit to one), and to waste your wealth". [Al-Bukhari and Muslim]

Al-Hafiz Ibn Hajar, may Allah have mercy on him, said:
His saying: "And the abundance of questioning" - does it refer to questioning about money, or questioning about problems and dilemmas, or is it broader than that? The first one is generally accepted.
Some scholars have already argued that there is a great question about the news of mankind and the events of time , or that of a certain human being . It was narrated by Abu Dawood from the hadith of Muawiyah. It has been reported from a number of the early generations that they disliked the excessive pursuit of questions that are usually impossible or very rare to occur. They disliked this because it involves pretentiousness and speaking based on conjecture, as the person engaging in it is not free from error.
"Fath al-Bari" (10/407).

And Al-Qurtubi, may Allah have mercy on him, said:
"And the point is: to take the hadith in its generality, so it encompasses all those aspects." "The Understanding" (5/164).

And Al-Nawawi, may Allah have mercy on him, said:
"As for (excessive questioning), it has been said that it refers to being overly persistent in matters and asking excessively about things that have not occurred and for which there is no need. Authentic hadiths have consistently warned against this, and the early generations disliked it, viewing it as an unnecessary burden. In the authentic texts, it is stated: 'The Messenger of Allah, peace be upon him, disliked questions and criticized them.'"
It was said: what is meant by this is asking people for their money and what is in their hands, and the authentic hadiths have consistently prohibited that.
It was said: It is possible that what is meant is the excessive questioning about people's news and the events of the time, and what does not concern a person. This is weak; because this has been known from the prohibition of "he said, she said."
It was said: It is possible that what is meant is the frequent questioning of a person about their condition and the details of their affairs, which falls under asking about matters that do not concern them. This includes the discomfort that may arise for the one being asked, as they might not be inclined to share their circumstances. If they do share, it may be burdensome for them; if they are met with disbelief or forced to imply something, it would also cause them distress. If they neglect to respond, it would be considered a breach of etiquette. "Muslim's Explanation" (12/11).

And thus, the meaning of "the abundance of questioning" is: asking people.

For more information on matters concerning which it is makrooh to ask, either in the sense of it being haram or discouraged, please see fatwa no. The hadith states: "Indeed, Allah, the Exalted, has forbidden you to be disrespectful to your mothers... and He dislikes for you to engage in idle talk, to ask too many questions, and to waste money."

And Allah knows best.