Here the english word desire is not enough.
Use taṇhā or chanda instead.
Arahants eradicated taṇhā, the cause of suffering, hence, they do not suffer mentally. Arahants still have chanda to be able to walk, eat, teach, sleep, go into the Jhānas, have compassion for the world etc.
taṇhā requires the ignorance of possibly seeing something with the value of possible permanent, unconditional happiness. Eg. likely the non-returner still might have some of it with regards to the form and formless realms, hence the fetters of attachment to form and formless realms are not yet eradicated for the non-returner.
chanda can still choose the relative better option of Jhāna vs staying in physical pain of the body. Despite knowing full well that ultimately both sides are dukkha of change and conditionality, but Jhānas lack the dukkha of unpleasant physical feeling. Note here that the arahant knows pain and just pain, without inserting a self in there (like Buddha’s advice to bahiya). There is no mental suffering corresponding to the physical pain, but he is still able to choose to go into Jhāna to get relief from the physical pain. Simply because the Jhānas has less dukkha.
chanda can still choose to act out of compassion, for recognising that the act of teaching produces happiness in the world, and may even lead people to the permanent happiness they have found. Compassion only is able to work by clearly discerning which outcome is less dukkha and which is not. So the overall dukkha of change and conditionality might play a role in the guide of compassion is to lead sentient beings from the conditioned to Nibbāna.
The dukkha dukkha of unpleasant feeling suffering also plays a role for compassion to recommend Jhānas compared to being in physical pain. Or worldly success, Being rich as opposed to being poor etc, to avoid hunger. So Buddha taught how lay people can be successful, living in the world. Sigalovada sutta, maṅgala sutta. etc.