Given that mitochondria have their own DNA and can replicate independently, can they ever become cancerous? For example, could a mutation in their DNA cause them to rapidly replicate, ultimately killing the cell it's in?
As a prelude, I should probably mention that single celled organisms cannot get cancer as we understand and define it. Mitochondria are not, of course, single celled organisms, they are organelles, but this interesting question involves treating them as if they were autonomous. We'll come back to that later. First, single-celled organisms and cancer...
Of all the hallmarks of cancer, unregulated replication is probably the most fundamental, but all the same, cancer is necessarily a disease of multicellular organisms with differentiated tissues. Bacteria do not get cancer. Protozoa do not get cancer. Yeast do not get cancer. I'm spending too much space on my prelude here, but let's pretend like this isn't an issue and just address unregulated replication. Incidentally, the review I linked is an updated version of the classic review of cancer biology. If you're interested in cancer biology, I highly recommend reading it, and perhaps the original 2000 review.
Ok, now to the question. Let's put aside the problem of a single organelle or single celled organism getting cancer, and consider whether mitochondria can develop mutations in their DNA that cause them to replicate in an unregulated fashion.
Yes, mitochondria do have their own DNA, but they are not fully autonomous. They do "replicate independently", but what is meant by this is that mitochondria replication (via fission) is a process that occurs independent from the replication (via mitosis) of its enclosing cell. The process is regulated by proteins encoded in the nuclear genome (not the mitochondrial genome)
Both fission and fusion are regulated by dynamin related proteins, which you can read about in this, fairly dense review. The mitochondrial genome encodes very few genes, 14 at last count, and while problems with those genes and proteins are implicated in a few diseases, uncontrolled replication is not one of them.