The menthol found in lip gel, balm, and even soap is a crystalline alcohol derived from mint oils thus shares the odor and properties of peppermint. Application stimulates your cold-sensing nerve endings, and the cool feeling is from the activation of those senses. (If you don't believe me, put on a mentholated product outdoors on a crisp day – you'll feel other cold sensations as even colder!)
In addition to soothing dry skin, it's also a common active ingredient in topical anesthetics and anti-itch creams since it relieves pains from burns and stings. Still, some people complain that products including menthol irritate their skin. Find out a couple possible reasons when you read more.
Since menthol's part of the alcohol family, it shares skin-drying properties common with other alcohol-based products. Some people with dry skin are more affected by this dehydration and find the results to be more chapped and irritated than soothed.
Another reason is that menthol quickly penetrates the skin's layers and can easily carry other chemical compounds with it. There's a chance that it's actually one of the other ingredients in the product causing the irritation, not the menthol itself.