When I was working on my project (asp.net mvc 3) before uploading it, I needed to make it refer to an already existing mysql database. There was/is no option of moving or cloning that database, and I had to add a block of code to my C:\Windows\Microsoft.NET\Framework64\v4.0.30319\Config\machine.config file for it to be made compatible.
How do I make it work on appharbor without moving my database?
You should be able to use mysqldump to move your data to a database running in the context of the AppHarbor MySQL add-on. Here's a guide.
If your current database is publicly accessible it should also work if you configure your AppHarbor app to use that. If you need to open a firewall, you can find AppHarbor app server IPs here.