Does the Quran instruct Muslims to kill non-Muslims?

There is no contradiction, praise be to Allah, between the statement that there is no compulsion in (acceptance of) the religion and the command to fight the polytheists. The command to fight the polytheists is not for the purpose of forcing them to enter Islam, otherwise the Jews, Christians and others would have been forced to enter Islam whenever the Muslims defeated them and they came under their authority. But it is well known to anyone who has the slightest knowledge of the history of Islam that this did not happen, because the Jews and Christians continued to live under the authority of the Muslim state, and they enjoyed religious freedom.

Rather what is meant by fighting is two things:

1. Fighting those who want to attack the Muslims in their own lands, and to spread the influence of disbelief and its followers in the Muslim lands. This is Jihad in defence of Muslim territory, and it is something that has occurred in every nation and state known to history, no matter what their religion, otherwise there would be no state and no authority at all.

2. Fighting those who bar people from the religion of Allah and prevent the Muslims from calling people to the religion of their Lord and spreading His light so that it may be seen by anyone who seeks guidance, or they prevent non-Muslims from learning about this religion or entering it if they want to. This is jihad in pursuit of the enemy. Both types of jihad are prescribed in Islamic teachings.

Ibn al-‘Arabi al-Maaliki (may Allah have mercy on him) said:

… The words of Allah, may He be exalted, {then kill the polytheists wherever you find them} [at-Tawbah 9:5], are general in meaning and apply to every polytheist, but the Sunnah specifies that it excludes those referred to above, namely women, children, monks and ordinary, insignificant people, as explained above. What remains are those who are in a state of war against the Muslims or are preparing to fight them and cause them harm. Thus it becomes clear that what is meant by the verse is: kill those polytheists who are fighting you.

(Ahkaam al-Qur’an, 4/177)

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:

What is meant by fighting is fighting those who fight us to prevent us from speading the religion of Allah, as Allah, may He be exalted, says:

Fight in the way of Allah those who fight you but do not transgress. Indeed. Allah does not like transgressors

Quran - 2:190

(interpretation of the meaning)

(Majmoo‘ al-Fataawa, 28/354)

Ensuring justice and showing kindness to non-Muslims who are not fighting Muslims

Concerning that, Allah, may He be exalted, says:

Allah does not forbid you to deal justly and kindly with those who fought not against you on account of religion and did not drive you out of your homes. Verily, Allah loves those who deal with equity.

Quran - 60:8

(interpretation of the meaning)

In other words, Allah does not forbid you to be kind, uphold ties, return favours and be fair towards the mushrikeen (polytheists), whether they are relatives and others, so long as they are not fighting you because of your religion or seeking to drive you out of your homes. So there is nothing wrong with you upholding ties with them, because upholding ties with them in this case does not involve anything that may lead to negative consequences.

(Tafsir al-Sa'di, p. 856)

Islam prohibits killing non-Muslims living under Muslim rule

It was narrated from ‘Abdullah ibn ‘Amr (may Allah be pleased with him) that the Prophet ﷺ (peace be upon him) said:

Whoever kills a mu‘ahid (a non-Muslim living under Muslim rule) will not smell the fragrance of Paradise, although its fragrance may be detected from a distance of forty years.

Sahih al-Bukhari, 3166

What is meant is one who has a deal with the Muslims, whether that is by paying jizyah or a peace treaty with the Muslim ruler or a guarantee of safety from a Muslim. (Fath al-Bari by Ibn Hajar, 12:259)