Is there a database with diseases and symptoms?
I imagine something like being able input symptoms and getting a list of appropriate diseases?
There's an App call Ada. It's only available for iOS. It's an IA based bot that ask you about your symptoms and gives you a diagnostic. It was pretty accurate with some regular illnesses. It's much more advanced than a "symptoms checker", it can for example give you opinions based on statistical data. Also keeps record of your symptoms and dates, and ask you after if you are feeling better or worse or if you are cured.
There's nothing that does what you asked as a medical doctor would do, because a software is always a software.
Anyway a good reference (there should be also a function similar to the one you asked for) can be found on http://reference.medscape.com/