Do Muslims have to keep contracts with non-Muslims?

The Quran's command is general

Elsewhere Allah says, "And fulfil every commitment. Indeed, the commitment is ever [that about which one will be] questioned" (Quran 17:34). Your tenancy agreement, phone contract, employment contract and business deals are all commitments you'll be asked about, whoever the other side is.

The Prophet's ﷺ example

• Business with non-Muslims: he bought food from a Jewish man on credit and left his armour as security (Sahih al-Bukhari 2068). That's a normal contract, honoured both ways.
• The Treaty of Hudaybiyyah: he signed a peace deal with the pagans of Mecca that seemed unfair to many Companions, and kept it to the letter until the Quraysh broke it themselves.
• Treaties in the Quran: Muslims were told to honour treaties with those who kept their side and didn't break them (Quran 9:4).

He ﷺ also counted betraying an agreement as a sign of hypocrisy, alongside lying and breaking promises (Sahih al-Bukhari 34).

What about the laws of the country?

Many scholars describe living in a Western country as a kind of agreement too: you're given safety and rights, and in return you keep the lawful rules. See Do I have to follow the laws of my country as a Muslim? Islamic law has always protected the rights of non-Muslims who live under agreements with Muslims as well. See How are non-Muslims treated under sharia law?

When a contract has a haram part

If a contract requires you to do something clearly haram, like paying interest or selling alcohol, the problem is the haram part, not the fact that the other side isn't Muslim. Avoid entering such contracts in the first place where you can. If you already signed one before becoming Muslim, don't just walk away and cheat the other party; speak to a scholar about the fairest way out. See Can I have a mortgage as a Muslim?