Which to use depends a little on the context. In the general case, as this is, then 'eat food that was' is better. You would use '...eat the food that...' if there was a choice - i.e. that there were two different sources, and you had a choice of one or the other. That would only normally apply in a specific case, which this is not.

to eat food without THE

In some sects of Islam, Muslims are forbidden to eat THE FOOD that was paid for by non-Muslims.

In this context the sentence should be " In some sects of Islam, Muslims are forbidden to eat THE FOOD that was paid by non-Muslims" . Article "THE" is used here in this sentence because of the addition of the clause " that was paid by non-Muslims" to it.