DOM-based vulnerabilities arise when a client-side script reads data from a controllable part of the DOM (for example, the URL) and processes this data in an unsafe way.
HTML5 storage manipulation arises when a script stores controllable data in the HTML5 storage of the web browser (either localStorage or sessionStorage). An attacker may be able to use this behavior to construct a URL that, if visited by another application user, will cause the user's browser to store attacker-controllable data.
Burp Suite automatically identifies this issue using static code analysis, which may lead to false positives that are not actually exploitable. The relevant code and execution paths should be reviewed to determine whether this vulnerability is indeed present, or whether mitigations are in place that would prevent exploitation.
The most effective way to avoid DOM-based HTML5 storage manipulation is not to place in HTML5 storage any data that originated from any untrusted source. If the desired functionality of the application means that this behavior is unavoidable, then defenses must be implemented within the client-side code to prevent malicious data from being stored.