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.
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.
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.
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.
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 :)
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.
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.
Interpreting the language of descriptions is left to the DM. To me it is clear that the intention was to hinder any divination spell that reveals information about magical properties, but this is my interpretation, so check with your DM.
Due to the magic aura, identify will show obfuscated or false information, but it will also detect that the item is under the magic aura spell. While you know that the information might be falsified, you cannot identify its true properties until you get rid of the aura (with dispel magic or letting its duration end).
The spell explicitly states that you can change whether the item appeares magical or not. Beyond that you can make it show that the magic belongs to a specific school of magic. This is mainly designed to fool detect magic. Because identify reveals the magic aura spell anyway, showing a specific ability or effect would be pointless.
Best Answer
No, Nystul's Magic Aura still works against truesight and detect magic
Part of what Nystul’s Magic Aura does is make “divination spells reveal false information about it.” Both Detect Magic and True Seeing are Divination spells, so they reveal false information – separately or in combination.
In the case of a creature whose truesight does not come from a spell, as in the example of a Gem of Seeing, this also doesn't work. Nystul’s Magic Aura does not create a "visual illusion" that truesight can detect – it specifically misdirects "divination spells", not visual senses.
Similarly, for a creature whose ability to detect magic is not spell-based, Nystul’s Magic Aura changes the way the target "appears to spells and magical effects [...] that detect magical auras".