Once a healthy biological male has reached adulthood, do his cells continue to divide by both mitosis and meiosis? Outline when each type of cell division might occur.

In order to remain healthy, his cells must continue to divide as old cells become worn out or damaged. His body will use both mitosis and meiosis to repair and replace them.
Meiosis only produces the sex cells. Meiosis that occurs in men is called spermatogenesis. The process is initiated when testosterone levels rise at the onset of puberty and FSH begins to stimulate the Sertoli cells that nurse germ cells in the testes. Sperms cells are replaced continuously.
All other cells use mitosis to divide. Unlike meiosis, which produces haploid cells that are genetically different from the parent cell, mitosis produces genetically similar diploid cells. Mitosis is initiated when signal pathways alert cells that exact copies are required. This happens at different rates throughout the body. A skin cell, for example, is typically replaced in less than a month, while a neuron will probably only divide once during adulthood.
http://www.uwyo.edu/wjm/repro/spermat.htm