How long after they remove a suspicious mole can they tell if you if it's cancerous?

Pathology processing. Takes time, includes preparation of the tissue, fixing the tissue, causing it to adhere to the slide, staining it with dyes, and now ubiquitous immunological markers before a pathologisty can read the slide, and render an opinion. If there is a question, this goes from 36 hours, to weeks.
Usually 2 days. Biopsy reports are given to the doctor, usually within two days or sooner, after the biopsy. Some of the difficult cases may require additional stains and consultations thus taking longer. Results of first diagnosis of cancer are usually called to the attending physician by the pathologist.
Derm path. 7-10 days depending on the laboratory.