I have a large, solid cedar chest that was made for me 5 years ago that sap has started to come out of on the inside of the chest. What are my options for dealing with this? I know very little about wood. How do I clean it and will it do it again?

First are you sure it's Cedar,sap very rarely is exuded from Cedar.What you can do is this,scrape off the sap and seal the inside.