By strict RAW, you can sense the presence of illusions of all kinds, but you can't actually see exactly where they are, or that they are illusions rather than just some kind of magic in your vicinity. From the description of Detect Magic:
For the duration, you sense the presence of magic within 30 feet of you.
Simple enough; if there's magic, you can sense it.
If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
If a creature or object is covered by an illusion, they're not a visible creature or object, and the illusion itself is neither a creature nor an object, so you can't see its aura. An illusion that is not being "borne" by a creature or object is likewise not a visible object or creature, so you can't see its aura, either.
Where things get messy is if there is a creature or object that "bears" an illusion but is still visible. However, I can't find any cases that would fall under this category. For example, you could use Minor Illusion to cover someone's head, but that is a freestanding illusion that happens to be around a creature, rather than actually being borne by that creature.
As far as whether you want Detect Magic to function by strict RAW in your game is concerned, given the way you've phrased this question I assume you're more worried about it being too powerful than the opposite, so this should suit you.
The flask is saved for when the DM asks you "do you have any glass containers" when you are trying to harvest a blood, poison or venom from vanquished monster.
People then try frantically to calculate how many health potions they have drank recently to convince the DM they have a lot of glassware that can be used to contain the valuable ingredients.
RAW there is no info on that - the bottle does not magically vanish. You end up with an "empty health potion flask" you can do as you wish. If you don't really note it down it can be assumed you just threw it away. If you do - you have put it in your backpack.
Best Answer
Excerpt from Detect Magic
Except from Sorcerer description
So obviously, while considering that 5e does not separate flavor text and official rules the RAW is yes. (DM's can decide how to handle it on an individual basis obviously)
As for Warlocks:
So not necessarily, however once you have been blessed with an ongoing magic power like darkvision things change.