Is it possible that the Quran contains any errors? Did Allah guard its writing, transcribing, etc, to ensure it was completely error-free? If errors are possible, to what extent might an error exist (i.e. could it contain only typos? Or might it contain a completely wrong/misunderstood teaching)? And are there any examples where some scholars believe an error may be present?
In the time the Qur'an was revealed to the Prophet Muhammad (SAWS), most people were illiterate. In fact, the Prophet (SAWS) himself could not read nor write!
Therefore, to preserve the Qur'an, the verses had to be memorized by heart. People who knew the Qur'an by heart were known as huffaz: they were highly respected members in Islamic society. [I could not find any verifiable sources, but the number of huffaz alive at the time of the Prophet (PBUH)'s death were in the tens.] It was not until much later that the Qur'an was fully compiled into the book we know of today. Despite the transcription of the Qur'an, people still memorize it to this day.
The Qur'an is known to be written in beautiful prose -- so beautiful that any [significant] modifications to the original Arabic can be identified. Furthermore, if a copy of the Qur'an is read by a hafiz (someone who knows the Qur'an by heart), he should be able to identify any errors that may have occurred in the transcription.
Unlike the Bible, any modifications to the Qur'an are severely frowned upon.
Also, as @AlUmmat mentioned, the Qur'an is guarded from manipulation or changes:
Allah protects the Qur'an from all errors. When the text is being written, it goes through many scholars to see if there are any errors before the Qur'an is copied. There are no errors in the Qur'an.
Allah says in Surat 15 Ayah 9:
By Quran I assume that the book Quran we have today. We have very high confidence that Quran as we have today is very close to what was revealed to the prophet by God.
Muslims believe that there has been no error in Quran as revealed to the prophet and that prophet made no mistake in exactly presenting it to the people as it was revealed to him. During the time of the prophet, he was present and was the main reference for Quran. There wasn't a compiled written copy during his time. I assume that this is not the part you are asking about. Your question is about how well what was stated by the prophet is preserved from his time up to today.
After him the history becomes a little bit complicated. Scholars agree that Ali (PBUH), unarguably the closest person to the prophet and the fourth Sunni Caliph and first Shia Imam, had a created a written copy by 6 moths after the prophet's death. But that was a private copy. The main reference for Quran among people were the people who memorized the whole Quran.
Until sometime during the first Caliph's reign, Muslims didn't feel a need for having a written collections of Quran. The memorizers were present and many and could be used as reference. However during a battle many of them died and as a result the first Caliph felt a need to have a written form in case other memorizers die. So a compiled version was created during the first Caliph's time and it passed on to the second Caliph.
However, the canonization of Quran only happened during the time of third Caliph. His rule starts 12 years after the death of the prophet, i.e. 644 A.D. He also destroyed all other codified versions of Quran. From that point we are rather certain that what we have is close to what was compiled at the time of third Caliph. Today we have copies dating back to 688 A.D.
There are issue regarding how well they collected and canonized it. There are stories about disagreement about what was part of Quran and what was not, even among the companions of the prophet.
There were also extreme groups like Kharijites who believed that large parts of what we have today as Quran are not part of it (for example, some of them believed that the whole Sura Yusuf was not part of Quran, mainly because of their extreme conservative beliefs that looked inconsistent with the topic of the Sura to them, and unfortunately even today some people advice against reading it for young people, mainly because they don't understand it. Really unfortunate because it is one of the literal wonders of classical story telling and as Quran says in the start of the Sura it is the best of stories "أَحْسَنَ الْقَصَصِ").
However, Muslims generally believe that canonization was done correctly. Sunni because of their belief in the four Caliphs, and Shia because of confirmation by Ali (PBUH) and Imams (descending from him). Also many of the prophet's close companions were alive and present when Quran was compiled and canonized and it was done under supervision of two of most famous memorizers of Quran. Any major/significant/important diversions would have been strongly objected to by at least some of them and would been noted in history. (There are some Shia that believe that Ali's (PBUH) version had differences because of some hadith attributed to him, but that is not a popular view among Shia scholars. Ali (PBUH) was the forth Caliph and we don't have any historical evidence that he tried to change the official version of Quran canonized during the third Caliph's reign.)
After the canonization there hasn't been any major disputes about Quran. Copies of the canonized version were spread over the Islamic world and is preserved through multiple copies over a very large area of land. We are quite certain to have the canonized copy. So if we accept that the Quran is preserved well in the 12 years following the prophet's death during the time of the first 3 caliphs (632-644), then we can say that Quran is preserved well.
However, it doesn't mean that what we have is completely error and typo free. There are 14 famous narrations of Quran and they have some differences that even change meaning in some cases. But the variations in meaning are not many nor major and for a text dating more than 1400 years is not surprising. There are books that discuss the differences in these 14 variations in detail. (Some apologists try to get around this issue and keep the belief that Quran is preserved exactly as it was revealed by prophet by arguments which look rather weak to me, e.g. some may say that all 14 versions are correct, etc. in any case, what we know for certain is that Quran has not been kept as a single undisputed written document from the time of the prophet till today, there are 14 versions with small differences today and there has been arguments about Quran's content during the 12 years that followed the prophet's death.)
In short, we have a very high confidence that Quran we have today is very close to what was revealed to the prophet.
I personally find these arguments regarding Quran has not been changed because of the verse mentioned in the other answers rather weak. The argument is that that verse 15:9 states that God protects Quran in a very particular sense, i.e. that Quran as a book among Muslims will be as it was revealed during prophet's time (independent of what Muslims do and whether they keep following his will and orders). So the arguments claim that to prove that Quran is exactly as it was revealed we only need to prove the authenticity of one verse. The authenticity of that verse is then established using historical arguments. (The authenticity of that verse itself needs to be confirmed through some other external way and cannot be based on itself, that would presuppose what is to be argued for and would be circular.)
I think this is a weak argument. The argument is based on a particular interpretation of the verse (to be fair, I should state that this interpretation is a very popular one, probably the most popular one among scholars). There are verses in Quran that implies that if even prophet (let alone Muslims) do not follow God's orders God may take away Quran (verse 17:86 in conjunction with other verses warning the prophet not to diverge from God's path, these are hypothetical verses and we don't believe they happened, my point is in the hypothetical case that they did then there would be such consequences and similar consequences may follow if Muslims diverge from God's path). So it seems reasonable that if Muslims did not follow God's orders he may take away it from them.
There are also theological arguments based on the fact that God must preserve Quran to guide people.
This is also a weak argument to me. God can guide whomever he want however he wants. He doesn't need to follow what some people might think he should.
The most troubling issue regarding these arguments for me is that exactly same things are claimed by Jews, and based on Quran we believe that they didn't preserve it as it was revealed by God.
Some Muslims can add that Islam is the last religion to their arguments but in reality Jews claim exactly the same thing. They don't believe that there is going to be a new religion after Judaism. And in any case these all rely on some beliefs that are supposed to be secondary to the authenticity of these texts.
In short, I haven't seen any philosophical or theological argument saying God preserves Quran which is not stated by Jews also.
Quran is a great gift that God gave to Muslims. Like other gifts God may take it away if Muslims don't understand its value and perform their duty, don't follow his orders, or diverge from his path. It is the responsibility of Muslims to perform their duty and preserve Quran as it is given to them.
There are no errors in the holy Quran, Allah is most merciful choose his messenger with the knowledge that his guidance would be relayed to all of us exactly as intended. There is no debate about this, the Quran is 100% perfect from start to finish from the moment Allah sent his message to the messenger, to the day all believers go to paradise.