You probably can
But is a pointless action since the object has to be visible in order for you to be able use your action to see the aura (emphasis mine).
Detect Magic: you can use your action to see a faint aura around any visible creature or object in the area that bears magic
There is also no indication that the aura can pass though solid objects, thus, in a case of a partially covered object the aura will be of no use to determine the shape.
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.
Best Answer
No, detect magic doesn't automatically detect spellcasters as magical
The Sage Advice Compendium addresses a related question:
Detect magic, like other game mechanics, operates by this same logic with regard to what is considered magical. The spellcasting abilities of creatures (innate or otherwise) are considered "the background magic that is part of [...] the physiology of many D&D creatures". Detect magic is designed to detect magical effects, not the background magic that suffuses creatures or the universe.
Chris Perkins confirms this sort of interpretation here:
Given that the question he's responding to asks about spellcasters in general, it seems clear that his response is not specific to wizards - he's suggesting that the spell doesn't automatically detect spellcasters simply due to their magical abilities.