I am running an old version of roundcube ( 1.2.3 ) on an old OS with php 5.6.29

If I install the latest 1.6.7 with the appropriate mysql database updates, I am curious if the 1.6.7 will find the old php 5.6.29 workable, or if there will be a hard stop because of an incompatibility.

Probably. You have to maintain public facing applications. Most likely there are 'remote exploits', that can be abused. Just run 'old' versions on new os in containers, so you can run multiple versions and upgrade in steps. It is also possible to run multiple php-fpm versions.

Remember if your polluting the internet, others are required to clean up your shit.