Can a tender lump in neck be malignant? I've heard that cancerous lumps don't cause any pain??

Low probability. Tender, red or painful lumps in the head and neck region most often represent inflammatory lesions like abscess or inflamed cyst or hair follicle etc. Pain is a symptom of nerve irritation either by inflammation or blunt/sharp trauma. If cancer invades a nerve it may also cause pain. In principle, any lump should be investigate.