Rush is also vacating his appointments on external trusts and committees, including the Basin Reserve Trust, Sky Stadium Trust and Wellington Water Committee, to focus on his treatment.
Wellington mayor Andy Foster said it was important that Rush focuses on his health and family right now.
"Sean is a valued member of the council, and we look forward to seeing him resume his rightful place around the decision-making table when he is ready."
Rush is thankful for the support shown by his council colleagues at this difficult time.
"I'll knock this off and be back to finish the triennium (the rest of his three-year term). In the meantime, I request privacy for me and my family so I can give this treatment its best shot."