Nausea and upset stomach are among the more common and simple side effects of Aleve, which is a brand name for naproxen. Drowsiness, dizziness, headaches and heartburn are other common side effects. Doctors often accept these minimal side effects in exchange for anti-inflammatory benefits, according to WebMD.
Liver disease is one of the more serious side effects that occurs with ongoing use of Aleve, according to WebMD. Bruising, bleeding and difficulty swallowing are other more significant side effects. Some people also have allergic reactions to Aleve, which may contribute to dizziness and breathing problems, as well as rashes. Dark urine, vomiting and abdominal pain may also present as side effects.