It will remain faint
...in some cases.
As noted in other answers, the minor illusion spell, as an example, says:
If a creature uses its action to examine the sound or image, the creature can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
The spell doesn't say any conditions in which it stops becoming faint to the creature.
Similarly, silent image says:
If a creature discerns the illusion for what it is, the creature can see through the image.
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:
- Seeing the same illusion in a different context
- The illusion moved
- The character got lost, thinks he's in a different room
- A character is in a labyrinth, leaves, and comes back
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:
The rules don’t account for every possible situation that might arise during a typical D&D session. For example, a player might want his or her character to hurl a brazier full of hot coals into a monster’s face. How you determine the outcome of this action is up to you.
One DM may decide to treat the brazier as an improvised weapon. Another DM may decide the way that particular paragraph continues:
You might tell the player to make a Strength check, while mentally setting the Difficulty Class (DC) at 15. If the Strength check is successful, you then determine how a face full of hot coals affects the monster. You might decide that it deals 1d4 fire damage and imposes disadvantage on the monster’s attack rolls until the end of its next turn. You roll the damage die (or let the player do it), and the game 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:
Bob the Bard: Hmm. Okay, so you're saying I can't tell if I've been
here before? So I'm lost? Fine. I open the door, what do I see?
DM: It's a 20x20 room, there's some dirty straw on the floor, some
broken crates and barrels on one wall.
Bob: Hmm. Could it be the same room?
DM: Make an investigation check, with advantage.
Bob: 7. 18!
DM: Looking carefully, you realize that this is the same room you were in before. Furthermore, looking at the far wall you remember that it is an illusion, and, in fact, it looks insubstantial, and past it you can see a passage, just as you could before.
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.
