Not all people with cheilitis who have tried antidepressants have had their lips helped by the antidepressant drugs. Some have, some haven't.
Antidepressants do many things in the body, not just purportedly affect levels of some brain chemicals.
If you are depressed anyway, and your doctor thinks it's a good idea, you might choose to take antidepressants, and see if your lips clear up too.
Antidepressants don't actually work a lot of the time for depressed people, and any particular type of antidepressant doesn't even work for a majority of people who take it. So, they are not a magic bullet.
They can have unwanted effects on the body and mind, too, so they aren't without risk.
NOW: If you have been depressed for a long time, regardless of the situation with your lips, you should try to get treatment for the depression.
Clinical Depression is highly linked with having a very disrupted and unhappy external life, naturally, but it is also linked to a poor immune system, blood vessel hardening, more heart attacks later in life, more risk of natural death later in life, permanent changes in important parts of the brain, having more temporary illnesses, aches and pains, and of course a much higher incidence of suicide later in life.
Depression is a biological phenomenon, as well as mental and emotional, and the biological effects of it can be extensive and devastating, and not all of these are reversible. It really does cause physical as well as mental scars.
However, drugs often don't do much for it. But that doesn't mean you shouldn't try antidepressant drugs. Try drugs, try nutrition, try counselling, try exercise, try socializing, try many things.