How do I give my application on shinyapps.io access to my remote database?

If your database is behind a firewall, here are the IP addresses you will want to whitelist.

54.204.34.9

54.204.36.75

54.204.37.78

34.203.76.245

3.217.214.132

34.197.152.155

Giving access to a database on Amazon:

If your database is on Amazon, you'll need to whitelist the IP addresses. To do that:

1. Log into your Amazon AWS console
2. If your database is in EC2-VPC, click on the VPC Dashboard under the Database section. If your database is in EC2-Classic, click the EC2 Dashboard under the Compute section.
3. Navigate to the Security Groups tab
4. Select the security group you want to allow access to
5. At the bottom, choose Inbound Rules and click Edit
6. Click Add another rule
7. Choose the type of connection you want to allow, the port range, and enter the IP addresses above. Note that you will have to add them as CIDR blocks - for instance - 54.204.34.9/32
8. Note that you will have to add a rule for each IP address, so follow steps 6-7 for each IP address given above. 
9. Once you've finished adding each IP address, click Save.