You might get away with it on BE 2010 however if you were running a newer BE version if the servername and drive letter for the location are going to change this would almost certainly require a new full .
BTW for any version of BE we won't have specifically tested your scenario. The concern I have for BE 2010 is that you would have to remember that to restore your data you would be looking at redirecting from the backup sets for the old servername to get the full and any older incrementals before recovering the newer incremental sets against the new servername. Over time this could become prone to human error and even accidental deletion of a critical backup set.
Also we do NOT recommend running one full backup and then doing incrementals for ever, as there is too great a probabilty that something may go wrong with at least one backup set in the chain causing issues for restore
As such best practice is run a new full anyway and plan to periodically run new fulls so that you don't have an massively long incremental chain.
