The answer to you question is no, hot peppers can not cause burn injuries. But that doesn't mean that wiping your nose with pepper juice won't hurt, because as you know it definitely will. Let me give you some background information.
Hot peppers are hot because they contain a compound called capsaicin. This is a very unique and special compound that binds to nerves in the mouth and nose and makes them fire. This firing is perceived by the brain as hot and/or pain. Thus from you perspective, it is sometimes hard to distinguish between actual heat - pain and just the capsaicin activating the nerves that are designed to sense these things. Interestingly, capsaicin is used in medicine as a treatment for certain types of pain and is used to help people that have discomfort from shingles.
In the future, if you don't have access to a question answering service like this one, and you have questions like this, it's never a bad idea to schedule an appointment with your primary care physician
. While in this case your burning nose from the pepper will end up being fine, you never know when your next question will be important to your health. Good luck.