If you're using someone else's copyrighted work for your t-shirt business without permission, you could be liable for copyright infringement. If your t-shirt designs are based on someone else's copyrighted work, you'll need to get written consent from the copyright owner before you can sell them.
To protect yourself legally, use only designs based on original works that you've created or for which you have permission to use. You can also seek out t-shirt design contests where the designers retain all rights to their work, allowing you to reproduce it as a t-shirt design.