Because our astronauts are likely to spend the rest of their lives on Mars it follows that they will pass away there as well. When that day comes, there will be a memorial service and cremation ceremony, like customs on Earth dictate.
After the first four people arrive on Mars, they will be joined by four more every two years. Together they will form a small but growing community of individuals that cover a range of specialties and functions. All activities will continue and developments will be able to continue in the case of a death or illness within the group.