There are many cases when user login but nothing is changed. Forum menu stays the same with [Login] and [Register] links, you are still not able to create a to post a reply. There are other cases, when all is good right after the login, but after a little navigation in forum pages you see your status is logout (not authorized). 

The problem comes from either "1. HTTP / HTTPS mix" or "2. Cache Plugin Conflict".

1. If your website protocol is https:// and you have an SSL certificate, first please check and make sure the links to your forum pages are not with http:// protocol, when you go to forum with different protocol it doesn't keep the login status. You should always use the main https:// protocol. Please edit menus and pages and find URLs with http:// to your forum, then change the URL. Make sure you don't have other links with http:// : This plugin may help you do it automatically: https://wordpress.org/plugins/http-https-remover/

Handle http:// to https:// redirection. Make sure your website URLs are https:// in Dashboard > Settings > Permalinks admin page. Then go to Dashboard > Forums > Settings > General Tab and press [Update Options] to reset forum protocol. Call your forum URL with http:// protocol. If it doesn't redirect to https:// protocol, you should use one of these plugins:

• https://wordpress.org/plugins/really-simple-ssl/
• https://wordpress.org/plugins/https-redirection/

2. If you have a cache plugin you should exclude forum home page from your cache plugin. wpForo has built-in cache system for the forum part and it should not be overlapped by other cache plugins. Please read this FAQ topic: https://wpforo.com/community/faq/wpforo-and-cache-plugins/