How does Burst use and store IP addresses? 

Burst Pro utilizes the IP address to identify the country information. This process is carried out locally in the plugin, with the IP being compared against a GeoIP database. It is important to note that no IP address is ever sent to a third party while using Burst – in fact, no information is ever sent to a third party. To add to that, the IP address is never saved to the database, it is just used to identify the country and we save that country information to the database.
Both Burst Pro and Burst Free utilize the IP address for blocking tracking, provided the IP has been added to the block list in the settings. The system simply compares the user IP to the list you have created, without storing any visitor IP.