The full answer
Some products can, particularly certain natural or wheat-derived ingredients
Some lip products use wheat germ oil or similar gluten-derived ingredients. Because lip products are applied directly to the mouth and can be ingested in small amounts throughout the day, this is a more reasonable thing to check than most other cosmetics.
What to look for
Many brands now offer gluten-free lip products, and checking the ingredient list for wheat-derived components (or looking for a gluten-free label) is the most reliable way to confirm before regular use.
Important context
Individual symptoms, reactions, medical histories, food products, and local labeling rules can vary. Community experiences are not medical evidence.
For personal medical decisions, diagnosis, testing, or treatment, consult a qualified healthcare professional.