I've been looking at some edge-case scenarios for illusion magic, and found myself stuck when trying to rule what would happen RAW with the following.
Multiple illusion spells rule that once a character has noticed the illusion's true nature (either through observation or trying to touch the illusion), the illusion becomes faint to them, and they can see through it freely.
With that said, here is the problematic scenario: let's say we have such an illusion of an object, and a character has noticed the illusion. The character can now clearly see through the object and knows at a glance that it isn't real. What would happen if that character lost sight of the object, then saw it later in a different context (maybe the objects around the illusion got moved around to make it seem like a different place, maybe the character got lost and thinks he is in a different room with the exact same layout)?
To put it simply, what if the illusion was previously identified as such by a character, but that character is now unaware that this same, previously identified illusion is in front of them?
Not having found a direct explanation in the rules of those illusion spells, I'm guessing this could go two ways:
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The character does not know it is the same illusion, so it appears real to them. They will have to identify it again in order to notice it is an illusion and see through it.
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The illusion has already been identified once, so it stays obvious to the character that it is an illusion. Thus, the character should know at a glance that this is the same illusion they identified before.
Depending on the way this works, this could make some heavy difference to some scenarios depending on illusions. For example, imagine that the character is in a labyrinth where one wall is replaced by an illusion. The character identifies the fake wall as an illusion, goes back to fetch the rest of his party, but loses his way in the labyrinth. Could they notice the illusion wall at a glance, despite not knowing where it is exactly anymore?
Best Answer
It will remain faint
...in some cases.
As noted in other answers, the minor illusion spell, as an example, says:
The spell doesn't say any conditions in which it stops becoming faint to the creature.
Similarly, silent image says:
However, that does not mean that all illusions, everywhere, in every circumstance, behave the same way.
The role of the DM
The scenarios you described:
are probably not really covered by the descriptions of illusion spells. For instance, minor illusion only lasts a minute, and silent image only lasts for 10 minutes and requires concentration as well. Probably that's not what's creating these illusions. It may not be reasonable to assume that the rules in these specific cases apply to all illusions everywhere. In particular, the spells don't account for the specific scenarios you're describing.
It's up to the DM to decide when the rules apply and when they don't. When examining the rules hypothetically, we can really only give a hypothetical answer. But the game comes alive when played with players and a DM.
The rules talk about when the rules don't cover something:
One DM may decide to treat the brazier as an improvised weapon. Another DM may decide the way that particular paragraph continues:
It is entirely reasonable that the DM make adjudications for situations not covered in the rules-as-written. Such adjudications are themselves rules-as-written.
The actual answer
It depends.
As you noted in the comments, there's a metaphysical question here: is the "faint" status of the illusion linked to the character's knowledge of the illusion, or if the illusion "keeps track" of whoever identified it, and changes shape accordingly for each observer. The rules don't say either way. Personally, to me it makes sense that there isn't a magical register of who's seen what illusion, but instead it's more like the real world. Sometimes you recognize things you've seen before, sometimes you don't, it depends on the circumstances. Sometimes something jogs your memory.
Here is an example:
Tying it all together
Where appropriate, you can very reasonably decide a character may be fooled all over again by an illusion. The rules don't provide a lot of guidance here, so you have to make your own determination. If a character knows that what they are seeing is the same as the thing they saw before, it's reasonable that they recognize the illusion again. If there is some reason for them to be confused on the subject, it's reasonable to that they may need to investigate the illusion again to perceive it as it is.