Yes, Wolverine's body functions very much like our own in regard to his basic physical processes. He will likely lose cells at the same rate as we do, when he is not in an accelerated regenerative state. But the curious nature of his cells makes his physical existence quite different than ours.
- possess the power to rapidly regenerate in seconds what would be days for humans
- can regenerate without needing new mass or energy for the process (?)
- can regenerate even if he is reduced to little more than a pile of burned adamantium on the ground, as long as any surviving organic tissue remains he will be completely regrown. (with a corresponding loss of memory)
- His cells function like super stem cells, able to take on whatever form is needed in his reconstructed body, so nerve cells and brain cells (which are almost never replaced in the human body) are able to be regrown as well.
So given the curious nature of his cellular process, would his cells die like normal cells?
In humans, our body can replace every cell in our bodies, (save nervous cells) in a period of seven years, give or take. We lose skin daily, stomach lining cells hourly, and our tongue can be scalded by hot coffee and recover in a week.
His body, given the amount of abuse it suffers, would likely perform in a fashion similar to our own EXCEPT where extraordinary stresses would require it.
- He does not lose brain cells to alcohol poisoning, so he rarely gets drunk.
- He might suffer a burned tongue from a sip of hot coffee, but his next drink will be with a tongue that has regenerated between each swig.
- minor physical inconveniences (hangnails, kidney stones, paper cuts) have no meaning in his existence and are likely repaired in the time it takes for us to acknowledge they happened in the first place.
- Even slow acting poisons are in a race with his body's ability to recover from injury
- He does not develop cancer, as every cell is his body perfectly replicates whichever form was in existence before, including the mutant link to the source of his powers.
Here are some other articles on the Scifi.StackExchange about Wolverine's healing abilities which offer links, images and resources for the detail-oriented.