Yes — a Tirthankar does eat and breathe while living in the human body.

In Jainism, a Tirthankar is a soul with a human body during that life, so ordinary bodily functions like breathing, hunger, and eating occur before kevalajñāna and before moksha. After nirvāṇa, when the soul becomes a Siddha, there is no body, so then there is no eating or breathing.

In short:

• While embodied as a human Tirthankar: yes, they eat and breathe.
• After liberation (Siddha state): no eating, no breathing.