Hi there, and welcome to the forum!
Does Anvil do automatic database backup and restore like heroku?
We take continuous backups, so behind the scenes we can do point-in-time recovery of your data within certain bounds. Right now that’s a service we offer to customers on a support plan. That said, ahem, watch this space - we’re going to be announcing something pretty soon 
Can I get a database dump?
Yep! You can get a CSV export of any table with the download button - or, if you want to be more flexible, you can do whatever migration you like with our API. (Probably easiest to do this with the Uplink, which lets you run code on your own computer, but still gives you access to all the Anvil APIs.)