Consider the following scenarios:
A user browses to a SharePoint or other web site using a non-IE browser, enters their forms-based authentication (FBA) credentials and checks the box to remember their account info, which creates a persistent cookie. They proceed to click on a link to open an Office document, which produces a prompt again for their credentials.
If the user switches to an IE browser, they do not receive the additional prompt.
A user browses to a SharePoint or other web site using IE to browse. This user enters their credentials in the form and proceeds to click on a link to open an Office document. They receive another FBA challenge for credentials.
If they then place the URL of the site into either the Local Intranet zone or the Trusted Site zone, they do not receive the additional challenge for credentials.
For Scenario 1, the reason for the challenge in the non-IE browser is that when Office requests the persistent, it is retrieved from IE's cookie jar. Other browsers do not write their cookies the same way or to the same location; therefore Office cannot retrieve those cookies.
For Scenario 2, the reason for the challenge is due to IE's Protected Mode. If the URL is resolved to an IE zone that has Protected Mode set, IE will not share the cookie with other applications, like Office.
For Scenario 1, the workaround is either to use IE to take advantage that Office can retrieve the persistent cookie, or (for non-IE browsers) the user will need to log in again when opening Office documents.
For Scenario 2, the workaround is to ether remove the checkbox for Protected Mode or move the URL of the site to a zone that does not have it set.