Google likely has algorithms to detect shortner use since it is often used by spammers and malware types to hide. Is there a blanket small negative for their use? Maybe, maybe not.
If users start linking in mass with the shortener from other websites where the backlinks do matter, do you want to risk losing a little of the juice sending them through a redirect?
Also if the target urls change because of forum or domain migration in the future you still have to maintain all these with proper redirects or you lose those backlinks. Even if you do all that correctly through a migration, you still end up with what is a chain of redirects overtime and that is definitely going to lose you some backlink juice. Google representatives have even said something to the effect of Googlebot gives up around 4-5 redirects in a chain. (I don't remember exact quote).
IMHO, potential false categorization by Google or potential small ranking algo negatives and long term redirect maintainence issues outweigh the small conveinence factor.
Also, why not edit the Forums item on the menu on the main site to point directly to
https://forums.servethehome.com/index.php instead of it having go through a redirect?