Once seen on TV, a woman interviewed on why she avoids having a biopsy for her suspected cancer, claiming that a biopsy will only accelerate it, thus more dangerous than beneficial.
Searching for a source of such a claim, I've found this:
Sadly many have been led to believe that early cancer screening will save their lives, nothing could be further from the truth. In fact biopsies often are instrumental in seeding the cancer elsewhere in the body. Thus accelerating cancer growth (metastasis) that was once contained by body and may have been slow progressing as is often the case. Clearly this is very dangerous yet most will subject themselves vily nily to this procedure in the misguided faith in the early detection myth... Any advantage one gets from early detection has now become a liability as cancer growth rate has accelerated often to the point that the patient has now become an emergency case.
Which also references this article:
Even though cancer is considered abnormal and rapid growth of cells, the entire mass of a cancer is normally rather slow growing -- because it is "encased." It grows inside a wall, or shell. The body is trying to protect itself from the cancer -- because cancer would otherwise spread throughout the body. So, the body tries to protect itself by building a wall around the cancer to keep any of its cells from "leaking out." (Actually, some cancers are very slow-growing [breast and prostate] while others are very fast [liver cancer].)
However these claims seem to be controversial, as doctors still doing biopsies routinely.