How can I stop Drupal from tracking or collecting IP addresses? Is there a way to stop Drupal from tracking or collecting IP addresses of visitors? The specific need case is for a site that wishes to be able to say no identifying information is kept of visitors (it sounds seedier than it actually is).

I recognize that Apache logs are going to do this, and that will need to be handled separately, but I'm wondering if there's any way to prevent this in Drupal (or if you have suggestions on how to obfuscate incoming visitor IPs, that would be helpful as well)
As mentioned in clives comment you could override $_SERVER['REMOTE_ADDR'] in your settings.php to obfuscate incoming visitor IPs. Drupal's ip_address() function than would return your obfuscated IP to any module that attempts to log IPs (like dblog).
Just be aware that any static or random address would conflict with a part of Drupal's flood control mechanism (the part that blocks IPs after too many failed login attempts; default: 50 within one hour). A static IP would expose you to the risk of locking every user out of the system. A random IP would render it useless.
Hashing the IP should work. The module cryptolog seems to to what you want and takes care of the flood control issue. From its project page:

Cryptolog enhances user privacy by logging ephemeral identifiers rather than actual client IP addresses in Drupal's database tables and syslog.
When Cryptolog is enabled, Drupal 8's \Drupal::request()-&gt;getClientIP() and Drupal 7's ip_address() output a 128-bit HMAC of the client IP address (in IPv6 notation), using a salt that is stored in memory and regenerated each day.
Because Cryptolog uses the same unique identifier per IP address for a 24-hour period, it is still possible to do some statistical analysis of the logs such as counting unique visitors per day. In addition, unlike other solutions of setting $_SERVER['REMOTE_ADDR'] to either one static IP address or a completely random IP address, Drupal's flood control mechanisms still function as normal.