Relics for better starter weapons are worthwhile on Revenant, and Executor. That's it.
Rev's claws are very nice to have even in the end game if you get them to purple, not just because of the revives, but a melee all faith fallback is hard to come by. Also Rev has the weakest level 1 in the game so starting with a hoarfrost or something is really worth.
Executor is worth because you can reliably get a double status frost/bleed with it and since status effects are basically the only reason to play Exec, it's worth. You can pick up whatever random blue you find on your run to sepuku yourself for the passive if you really want, then go back to your starter to attack with.
