Eldritch Sight (PHB 110) allows you

cast detect magic without expending a spell slot

An illusion is magic, so they will detect its presence automatically within 30', and will also be able to use an action to see the aura around the specific source and determine the school of magic, per the Detect Magic spell. Note, the illusion itself won't show an aura, but the trap trigger should, if the warlock is within range of it when it is triggered.

According to Jeremy Crawford (lead rules developer on 5e), when he was asked on Twitter

Would Detect Magic show an aura around an illusion?

His response was

Detect Magic lets you see an aura only around a visible creature or object, not around an illusion. But you can sense the magic.

So the warlock would be able to sense the presence of the magical trigger or illusion source and could detect that it has an illusion school aura if he can spend an action. If the trap is already triggered (assuming the illusions are cast by a trap) then the illusion itself will set off the magic "spidey sense" but wouldn't have an aura. The warlock would have to be within 30' of the trigger/source.

There are a few ways to confound a player using Detect Magic, though you really ought to accept this PCs capability and use it for good story moments, rather than trying to shut it down, which will only frustrate the player.

1. There are so many things radiating with magic that the PC can't distinguish between illusions and just ordinary things that have some residual magic aura without spending a lot of time (actions). Presumably whoever created this haunted house knew of Warlocks and this ability, so they planned accordingly. So give the PC lots of things to investigate, eventually they will get bored and walk into a trap. It is also possible that the use of Nystul's Magic Aura could mask the magical aura of the trigger/source as another school of magic. I would postulate that a warlock who uses Detect Magic frequently would habituate to the magic used by the party (the fighter's magic sword, the thief's magic ring, etc) and could detect a new magic source, lest Detect Magic become fairly useless unless the warlock is alone and divests himself of all his other magic items.

2. Using this magical ability attracts negative attention. Maybe the house responds aggressively to use of magic, the walls close in, things fly at the magic user, etc. This will force the PC to limit use of the ability.

3. Accept it and let the PC detect all the illusions and their source if he is close enough. But remember that things can block Detect Magic, like 1 foot of stone, 1 inch of metal, etc. So chests and such could be behind secret, non-magical doors.

4. The range is also only 30', so things farther than that won't show an aura. An illusion on a high ceiling, or at the end of a long room will require the PC to get close to even tell there is magic involved. Use that to your advantage to lure them into non-magical traps. After a few, they will learn to stay back :)

5. This is also eating up their concentration slot, as the invocation lets them cast the spell without using a slot but doesn't relive them of the need for concentration. So they can't have another concentration spell active. And if something happens that requires them to make a concentration roll and they fail, there could be a brief period of time where an illusion could pop up and the warlock won't be able to sense magic, or determine that it is just an illusion. So you could have a mundane threat that injures the warlock, breaks their concentration, then you trigger an illusion that they do react to. Your warlock may forget to recast the Detect Magic spell or prioritize other actions instead.

6. Mundane creatures with magic items will also trigger the magic sense, but the warlock will have to burn an action to see the source. Have a series of skeletons come at them, the first two just illusions, by the third the warlock may make an assumption that it too is just an illusion and then someone gets hit with the magic sword a real skeleton is carrying! Or vice versa, so the party wastes resources attempting to attack the third illusionary skeleton after fighting the first 2 real ones.

But again, you should embrace this ability, as it is the PCs area to shine and something they obviously care about, since they took that invocation over other ones. Be careful trying to negate it completely. Instead use it to make the character take risks.

They are "real images"

Illusions can create either a "real image" or sensory experience that anyone can perceive, or plant images directly in the mind of creatures.

The Schools of Magic, PHB 203

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.

This implies that "phantom images" created such that any creatures can see them (Silent Image, Programmed Illusion), are distinct from illusions that plant images directly in the mind of a creature (Fear, Phantasmal Force). This further implies that the images created by silent image are not planted directly in the mind of its observers.

Mechanical Perspective

The spells minor illusion, silent image, programmed illusion, and disguise self all have a range.

Let's take for example, the spell disguise self. It has a range of Self and can change the appearance of the caster superficially. These magical alterations, though limited only to the caster (as seen in the range of the spell), affects the perspective of anyone who can see it -- even from hundreds of feet away, if visibility allows.

Similarly, the spell silent image can create the image of an object within 60 feet of the caster, affecting an area no larger than a 15-foot cube by creating a visual phenomenon in that space. Therefore, the spell does not affect the minds of its observers (at least, not directly), but rather only a specific spot within range of the caster.

So, if someone 300 feet away looks at this image, they should still be able to see it, since they aren't being directly influenced by the spell. They're just perceiving a space differently, and that space is what this spell is affecting.

The same arguments can be used for programmed illusion and minor illusion. These spells create illusions that exist outside the mind of their observers.

The "paradox" of different people seeing different things

Once a visual illusion is discerned, it becomes faint to that creature and to that creature only. Meanwhile, others continue to see it as if it was really there. This imparts to these illusions properties of a mass hallucination, because two people can look at the same place and see different images (one sees a "full" image, the other sees a transparent one).

Strictly speaking, from a mechanical PoV, these images are external to the observers. So there should be an in-world explanation for why they have properties that suggest they are hallucinations instead.

This can be explained any number of ways, though, without being cornered into accepting the illusions as directly affecting the mind of the observers. First of all, it's magic, which means it doesn't have to be intuitive. One way to explain it is, since these illusions affect a space (not you directly), and you are just perceiving that spot differently, the Rules of Magic could change depending on your knowledge as the observer. There are other spells which have this property too: zone of truth does not allow you to deliberately lie, but you can say something that is false if you believe it to be true; geas could allow you to act against the direct instructions of the caster if you don't know you are violating those instructions (otherwise you could use geas to say "Say only things that are true about X" and everything the target says will have to be factually true about X, even if they had no way of knowing if it was true otherwise).

Another way, which is simpler, is to say the spell could be presenting you with an optical illusion, one whose nature is such that once you see it, it's hard to "unsee."

Implications

Antimagic Field

The spell says:

Spells. Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.

This clearly means illusions that create images in the minds of its targets are absolutely nullified if the target enters an area affected by antimagic field. But they can still see an image (created by, say, silent image) that is not in that area, for ex., one just standing outside the border, because that image is not inside the area affected by antimagic field.

Detect Magic

The spell says:

For the duration, you sense the presence of magic within 30 feet of you.

If a magical illusion is within 30 ft of the caster, then the caster will detect it (but they won't necessarily know where it is, or which school it's from). If a phantom image that anyone can see enters the range of 30 ft, then they will sense magic is present. If someone affected by the fear spell enters their range, they will also detect the presence of magic.

Light

This is a messy topic, and is best left to DM adjudication.

If an illusion cannot block light, then it must be invisible -- you would see through it because, to see in the first place, you need to see the light coming off it. If it can't block light, then that means you are still seeing the light coming from the object directly behind it, and the illusion is covering nothing.

On the other hand, if an illusion can block light, then illusion spells which say: "[p]hysical interaction with the image reveals it to be an illusion, because things can pass through it" are contradicted, because light is a thing. If light, as a thing, can't pass through the illusion, then the clause on illusions being unable to block things is violated.

So the implications for light, as far as illusions go regarding "are they visible to all, or only in your head?" are messier and tangentially related at best. It is probably applicable only to spells which conjure phantom images that anyone can see, though, and not to illusions that plant images directly in your mind.

Vision

Line of sight is defined in the DMG:

Line of Sight, DMG 251

To precisely determine whether there is line of sight between two spaces, pick a corner of one space and trace an imaginary line from that corner to any part of another space. If at least one such line doesn't pass through or touch an object or effect that blocks vision -- such as a stone wall, a thick curtain, or a dense cloud of fog -- then there is line of sight.

If something blocks your vision, you do not have line of sight. If an illusion obstructs your vision -- ie, you cannot see what is behind that illusion -- then your line of sight is cut.

Notably, even a thick curtain can block line of sight. Thick curtains probably cannot shield you from an arrow or a beam created by disintegrate, but it can obstruct the vision of those who would use these things against you, and that is enough to stop them from targeting you.