Can the intangible illusory copies obtained by casting mislead move through walls?
If the illusory copies cannot move through walls, what would happen if an illusory copy tries to push someone else?
From the rules text:
the invisibility ends if you attack or cast a spell (PHB. p. 260)
I am making an interpretation here, because the character controls what the illusion does, and "you attack" seems to fit with "the illusion attacks" depending upon whether you are moving your illusory double, or if you have switched back to your own senses. It's unclear if the illusion can attempt an attack at all. A push fits into the category of shove which is a special form of melee attack. (p. 195, PHB, Shoving a Creature).
Using an attack action, you can make a special melee attack to shove a creature ...
Moving through walls is not addressed in the rules, so there you have to get a ruling based on either "if it doesn't say I can, I can't" or "if it doesn't say I can't, I can" which is as much a DM/table philosophy as anything else.
Since the general intent in D&D 5e is that "spells do what they say they do" it's a fair ruling to say "No, the illusion created by mislead doesn't move through walls."
Moving through walls, if allowed, would likely give away the illusion to any other observer, so as DM it would be appropriate for anyone seeing the mislead-created illusion moving through walls to have advantage on an ability check to notice that it is an illusion, or to recognize it as an illusion of your character without a roll.
(The comparison to Project Image has been removed now that the question has been updated).
Mislead tells you it can co-occupy and pass through occupied spaces in its first sentence, which reads (emphasis added):
You become invisible at the same time that an illusory double of you appears where you are standing.
It literally appears where the caster is standing on top of their (now invisible) body.
Supplementally, all spells that require sight carefully include the wording "you can see". For example, contrast the wording of the Acid Splash cantrip (emphasis added):
Choose one creature you can see within range, or choose two creatures you can see within range that are within 5 feet of each other.
with the wording of the Chill Touch cantrip:
You create a ghostly, skeletal hand in the space of a creature within range.
Note that moving the illusory double created by Mislead does not contain the "you can see" wording:
You can use your action to move your illusory double up to twice your speed and make it gesture, speak, and behave in whatever way you choose.
The description of the illusion school of magic in the PHB/basic rules says:
Illusion spells deceive the senses or minds of others. They cause people to see things that are not there, to miss things that are there, to hear phantom noises, or to remember things that never happened. Some illusions create phantom images that any creature can see, but the most insidious illusions plant an image directly in the mind of a creature.
The double being illusory as stated in the spell, it is not real. As you would expect from illusions, they don't actually interact physically with the world. In other words, it would not be blocked by a wall or even a creature for that matter. Same applies to Major Image, for example.
However, as pointed out by KorvinStarmast, witnessing it passing through objects or creatures would undoubtedly give away it's illusory nature. Shoving a creature might feel like a bump... but all the implications of a bump have no reason to physically happen.
In many illusion spells a lot is left unsaid; even more so in Mislead. This is on purpose. Jeremy explicitly said so in this Dragon Talk: Sage Advice on Illusions. The intent is to allow the DM to adjudicate the limits of illusions and encourage players to come up with creative uses. Notice that Jeremy mentioned different examples of illusions passing through barriers or objects/creatures passing through them. That is because the assumption is that unless said otherwise, illusions aren't actually physical.
The question begins, "Can the intangible illusory copies obtained by casting mislead move through walls?"
The querent is assuming that mislead creates intangible illusory copies. In fact, the spell does not say that the "illusory double" is intangible. In contrast, the "illusory copy" of yourself created by project image is explicitly said to be intangible (and to be fair the original version of this question asked about both mislead and project image).
To be clear, I do believe that the misleading double is intangible, but it is important to distinguish between RAW and interpretation, and mislead does not actually specify whether the double is tangible or not. Let's assume from here on that it is intangible.
Our question then becomes, "Can the illusory copies move through walls?" And while it might seem obvious that intangible things can pass through a solid wall, that too is interpretation rather than RAW. In fact, it is not actually clear what "intangible" means. To be more precise, the rules do not specify how an intangible things interacts differently with the world than a tangible one. Contrast this, for example, with incorporeal, which has a clear and consistent effect on movement and damage.
'Intangible' is used once in the Monster Manual (for the illusory self-image of an aboleth), once in the Dungeon Master's Guide (in flavor text quoting a Richard Baker story), and thrice in the Player's Handbook (in the spell descriptions of clairvoyance, imprisonment, and the aforementioned project image). Nowhere is is defined in a game sense, and nowhere does it say that intangible things can move through solid objects.
So, rather than worrying too much about 'intangible', let's look at the nature of illusions. Many illusion spells conclude with some variation of 'Physical interaction with the [illusion] reveals it to be an illusion, because things can pass through it' in their description. Mislead does not use this phrase, although I believe that this was an editorial oversight. So, as with intangible, let's proceed assuming that mislead should have this standard text about things being able to pass through it.
It seems both logical and reasonable that if solid things can pass through the illusion, then the illusion can pass through solid things - and yet in a strict RAW sense, nowhere does it ever say that. Further, for all the illusion spells that allow one to create movable images, nowhere does it ever explicitly say they can be moved through solid objects like walls.
Many illusion spells say that their images appear at 'a spot within range'. This is notably different from creature-summoning spells, which typically say that the creatures appear in 'unoccupied spaces that you can see'. The fact that illusory images are not restricted to spaces that are unoccupied is another clue that they can, in fact, pass through solid things. It could be argued that the ability to place an illusion on 'a spot within range' rather than 'a spot within range that you can see' implies that you could start your illusion on the far side of a wall or inside a solid object. However, the counter-argument to that are the "clear path to the target" rules, which say that even if you don't need to see the point in space you are targeting, it still cannot be behind total cover to you. Thus you could not make a silent image of an illusory ghost starting inside a wall and later have it move out.
As user2105464 points out in their answer, the description of mislead explicitly states that the double appears in an occupied space - the space that you are occupying! This doesn't confirm that it can move through such a space, but it is pretty strong evidence that it can.
Finally, consider the description of major image. There, the illusory image you create "appears at a spot that you can see within range". Once you have made it, though, you can "use your action to cause the image to move to any other spot within range" without regard to whether or not you can see the spot. This strongly implies that it would be possible to create the image on the near side of the wall, and then have it pass through to the far side. If that is a working example of how illusions function, it seems like your misleading illusory copy can be sent through walls as well.
RAW, nothing says that the misleading copy is intangible or can move through walls, and in fact no illusions are explicitly said to be able to pass through physical objects. However, the illusion in mislead appears in an occupied square and similar illusion spells permit physical objects to move through them, so a reasonable interpretation is that these kinds of illusion spells, mislead included, can move through walls and other solid spaces.
It should be clear from the above that I think the misleading copy is insubstantial, and thus incapable of a physical attack. As evidence for this, 'spells do only what they say they do', and nowhere in the spell description does it say that the illusory copy can attack. This might appear to be special pleading since my argument here is that 'mislead doesn't say it is intangible or can be detected, but I bet it was intended to and that was left out as an editorial error.' How can I be confident that the error was not actually in leaving out the part about using the copy to attack?
The difference for me lies in the practicality of adjudicating the flawed spell. If we assume that the illusory copy was meant to be intangible but detectable, we can default to the 'pass through' detection rules used in so many other illusion spells. But if we assume that the illusory copy was meant to be a creature that can attack, we still don't have a stat block for it. If we assume the illusory copy was supposed to be an animated object, we still don't know its AC or hp. Granting the spell the capacity to attack doesn't tell you anything about how the spell attacks, and there is no standard illusory stat block to default to.
However, suppose a DM rules that the misleading copy can attack, such as by making a Shove, and assigns it a Strength score. Fair enough. In such a case I would disagree with KorvinStarmast's interpretation that the attack would end the caster's invisibility. Mislead says 'the invisibility ends if you attack or cast a spell' and your illusory copy is not you, any more than your familiar or your summoned animals are you (and you don't become visible when they attack, either).