What do we call a relationship where one organism feeds off another without killing it?

The relationship where one organism feeds off another without killing it is referred to as parasitism. In a parasitic relationship, the parasite derives benefits at the expense of the host, which continues to live but may suffer from various consequences such as nutrition loss, disease, or stress. For example, tapeworms live in the intestines of their hosts, absorbing nutrients from the host's food while the host remains alive.

In contrast, predation involves one organism killing another for food, which does not apply here. Commensalism is a relationship where one organism benefits while the other is neither helped nor harmed, and amensalism refers to a relationship where one organism is inhibited or destroyed while the other remains unaffected. Thus, the definition of parasitism accurately captures the nature of the interaction described in the question.