I'm 20 years old and I've been wearing contacts all my life. Will I have dry eyes when I'm older?

Maybe. Contact lens wear can cause eye surface irritation and dry eye symptoms. Dry eye is very common. I would try artificial tears frequently during the day, ointment at night, warm compresses 1-2 times a day, and omega-3 fatty acid supplements. If no improvement, see an eye doctor.
It is quite likely. Most individuals develop some dry eyes as they age. Genetics, medications and diseases can increase the dryness or likelihood of dry eyes. Contact lenses are sponges. As the eye remains open and water evaporates off the lens surface the contact draws water from the eye, making it somewhat dry. Contacts can irritate the eye, which can also contribute to creating dry eyes over years.