What does the Quran state about how to treat Jews and Christians in the following verse? {For you is your religion, and for me is my religion.} [Quran 109:6]

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger. 

This verse indicates disassociation from disbelief in all its forms. Ibn Katheer said, "This Soorah is the Soorah of disassociation from the deeds of the polytheists. It commands sincerity in observing this obligation. The Words of Allah, The Exalted: {Say: O disbelievers} [Quran 109:1] includes each and every disbeliever on the face of the earth ... This is why the Prophet, sallallahu ‘alayhi wa sallam, said to them: {For you is your religion, and for me is my religion.} [Quran 109:6] Allah, The Exalted, also says: {And if they deny you (O Muhammad) then say: For me are my deeds, and for you are your deeds. You are disassociated from what I do, and I am disassociated from what you do.} [Quran 10:41] He further says: {...For us are our deeds, and for you your deeds...} [Quran 42:15]

As for the way in which one is to treat the different types of non-Muslims after fulfilling the obligation of disassociating from them in terms of religion and beliefs, it is governed by the Sharia rulings which differentiate between Thimmis (non-Muslims living in the Muslim state under the protection of Muslims) on the one hand and the belligerent non-Muslims (who are at war with Muslims) and those who support them against Muslims on the other. Allah, The Exalted, says (what means): {Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly. Allah only forbids you from those who fight you because of religion and expel you from your homes and aid in your expulsion - that you make allies of them. And whoever makes allies of them, then it is those who are the wrongdoers.} [Quran 60:8-9]

For more benefit, please refer to fatwa 88293.

Allah knows best.