As cells divide, they accumulate mutations that can sometimes cause cancer. Gametes have to divide like any other cell, and thus generation to generation mutations should accumulate in people's genomes. However, it doesn't appear to be the case that cancer rates increase as a species ages. Why isn't this the case? Is the process of development itself a filter that ensures that most of the cells in the body are "far from cancer"?
However, it doesn't appear to be the case that cancer rates increase as a species ages
Species don't have an age. All lineages on earth are about 3.5 billions years old. No, lineage is younger or older than that. A lineage does not get its genetics magically reset when the lineage gets a new species name. For more information about the standard misunderstandings about the concept of species, please have a look at the post How could humans have interbred with Neanderthals if we're a different species?
In all cases, there is some equilibrium between the selection pressures brought about mutations that are linked to cancer (whether mutations on oncogenes or on tumor suppressor genes).
Individual, do have an age though. Cells, within multicellular individual replicate in order to ensure the multicellular individual growth and maintenance. Those cell lineage can accumulate mutations that could be linked to cancer development. Within the individual, these mutations are not deleterious and are therefore not selected against. Actually, in case of a cancer, those mutations are, by definition, beneficial as cancerous cells proliferate (have a large fitness). As a result one should expect that old organisms are at greater risk of developing cancer than younger organisms. In humans, this correlation exists, however, the statistics are slightly more complex than that. Typically, different types of cancer seem to show different age associations. See this article from the NIH as a start for more information.