Are there symptoms of oral cancer that show up on other areas of your body?

Metastasis. Oral cancer can metastasize to other areas of the body including lymph nodes, liver, lungs, bone. Oral cancer should be managed by ENT surgeons. Oral & maxillofacial surgeons in some regions.
Yes in Late Disease. Oral cancers start out small and grow. They generally grow into the surrounding tissue and cause local damage and ulceration at first. They can spread (metastasize), but that usually happens with advanced disease and is a very poor sign.