Can nasal polyps turn cancerous?

Almost never. True nasal polyps are inflammatory tissues arising from the lateral wall of the nose. They almost never degenerate into malignant tumors.
Rare. Inverting pappilomas, which can mimic polyps can turn to cancer. Inflammatory polyps do not tend to turn cancerous.
Can be a problem. Nasal polyps should always be further evaluated and monitored by a specialist (eg., ent) and surgically removed if needed. They can be cancer and should not be ignored.