Load balancing Sharepoint - constant logon prompts

Considering that a) you don't appear to be using APM, and b) it works for some browsers and not others, I'm going to assume that you're doing NTLM authentication to the SharePoint server through the LTM VIP.

If that's true, disable the OneConnect profile, which can cause issues with NTLM.

Also, assuming NTLM authentication, your browser has to be configured to support "Windows Integrated Logon". In IE, go to the Tools Menu -> Internet Options -> Security Tab, click on Local Intranet, then click the Sites button, and then click the Advanced button. In your IE browser you should a host name or wildcard host entry that matches your site. This is what allows IE to pass the user's logon token to a remote server that requests authentication.