What would happen after their deaths?

The film does not answer this, and no confirmation by production to date exists. They do say they left it open ended on purpose:

We had a longer ending with Andy Garcia walking out of the elevator. 'Why is he in one shot?' Because it was two scenes that we shot with him, but we find out that by doing the ending a little shorter, it made people talk more about it. I want to like, 'Did they have children? What happened?' It's good. Somebody will go like, 'Oh, I think I saw some children inside the house.' Somebody goes like, 'Wait, if you saw that, then there probably is!' 'How was their life?'

But they both get to do what they set off to do. She set off to write a story. She thought it was different story. She had to, instead of looking outward, she had to look inward... Which I think is amazing, and he build his house, and was able to live in it... So, in many ways, they completed what they needed, and the rest I want people to imagine and talk about and it should be up to them.

The original screenplay and drafts answer this in a different way.

Instead of saving everybody, the ship reboots and ejects every hibernation pod in the middle of space. Jim and Aurora are the only ones alive. The Homestead corporation had, as part of standard procedure, taken sperm and egg samples from each colonist. Jim and Aurora find this, and 80 years later when the Avalon arrives at Homestead II, which already has colonist there, the ship arrives, carrying adults and children. Implied to be created by Jim and Aurora using the frozen sperm and eggs. (I guess the medical bay had an Invitro option? IIRC it also had a pregnancy delivery option when we see the override screen.)

We then, of course, see a similar scene to that in the final movie, with a century’s habitation overgrown through the ship. We also get a glimpse of Arthur slicing vegetables at the bar, and a high wall in the concourse listing all the births, deaths, marriages, catastrophes and achievements of the last century of space travel, alongside a table with a collection of artifacts from the story including Gus’s worn shipcard, a book Aurora has written about the experience.

Alternative ending aside, the logic answer is that they weren't buried. Jim and Aurora are likely two decomposing corpses somewhere on the ship, for the crew or passengers to find. I'll hope they died together, fingers wrapped.

Remember Jim sent a message back to earth when he woke up! He was due to receive a response in 55 years from then. When that response arrived, it told him how to put himself back to sleep in his pod; He put Arora back to sleep in the auto doc, then put himself back to sleep in his pod. So they completed their journey to Homestead II. Of course, if they had kids along the way, Jim, being an engineer would have built more auto docs or pods to put their kids into so they could fall asleep too. That's my take on the ending.