More specifically, if a human subject was exposed to, say, a human cancerous cell line (via intravenous injection or through an open wound, for example), is it possible that they would develop any conditions associated with the cell line?
This is highly unlikely because the immune system would detect these cells and atack them as foreign. There is at least one case described in the literature in which a woman died of lung cancer after receiving a donor organ from a smoker. This is due to the fact that after transplantations people have to take drugs to suppress their immune system to avoid an immune reaction against the transplanted organ. This allows tumor cells to survive.
The same problem is true in research animals. For the transplantation of tumor cells you need mice with a severely deficient immune system (so called "SCID" (Severe Combined Immunodeficiency) mice) to allow tumor cells to grow. This is different when you work with viruses (like the human papilloma virus) which can cause cancer. Here you have to take appropriate safety measures.