So.... I am emptying out my ingredients. I come to my briar hearts, and - there are 4 in my inventory - and they tell me quest items cannot be removed!? What the heck?! I don't see any open quests that require them from what I see. I had done the first half of the White Phial.... And received word to go back for the second half. Will this resolve at the finish of the White Phial - or are my hearts permanent in my inventory? TNX!

Eat all the briar hearts and continue the quest. Make sure you don't pick up any more hearts until you take the quest related one.

That's got to be it. I get the same situation with Amulets of Talos until I do that quest.

On two characters, the second part of that quest is bugged and refuses to start. Im not to bothered about it though, it seems that I misinterpreted what the White Phial could do. I thought you would select a potion that was in you inventory and it would copy that potion. (a Potion of Ultimate Healing for my warrior (heals 100% health) or a custom Potion of Invisibility (about 130 seconds) for my thief/assassin) In reality, you choose from a list of effects when he hands it to you, none were of any intrest to me.

Yup. You need a Briar Heart for the second White Phial quest. Also, once an item becomes a quest item that cannot be removed from inventory, it does not weigh anything. You can test this by storing away everything that can be dropped and checking what your carry capacity is.

It won't start until you've completed "Throat of the World" in the main quest, which may account for the (seemingly) unreasonable delay in starting "Repairing the Phial" after finishing "The White Phial." Have you gotten that far in the main quest on both those characters?

(May sound like a stupid question but, well, I've had people report that quest as "bugged" before when they just hadn't gotten the necessary prerequisites. It happens.)

This. ^

It never started for me and then as soon as completed Throat of the World the message for Repairing the White Phial popped up.