Can you make it so Jotform does not keep logs to keep messages anonymous?

This is for an anonymous crime tip hotline website.  

You can use the Auto Delete Submissions widget: https://apps.jotform.com/app/auto_delete_submissions 

Let us know if you have more questions.

Understood that there is a plugin that removes from your system. My question is more specific, do you records of submissions with information such as ip information internally? Are there logs showing the IP information if there is a court order?

All submissions done on your form will be logged and stored inside your account through the Submissions Page.

Related guide: How-to-View-Form-Submissions

It includes all the form fields filled out by your users/respondents, along with their IP address. Now, those records will only stay there if and only if you let them stay there.

When deleting submissions, you will see this warning message (see screenshot below). Deleting submission data is irreversible and will permanently remove it in our servers, including the file uploads on them.

Related guide: How-to-Delete-Form-Submission-Data 

So, if you're concerned about anonymity, then deleting the submission will completely remove the logs/records from our servers, regardless if you delete them manually or if you'll use the app suggested by my colleague.

Is there any way to not keep track of the IP addresses to begin with?
...

The moment a form is submitted, your user's info will be logged, along with his IP address. If you'll use the Auto-Delete App, then there's no way for you (or for us) to know their submission data and IP.

Your only source of info would be the Email Notifications you'll receive if you have setup one. And the email template doesn't include the IP address of the respondent by default, it will only include the form fields.