So one of my players decided to play a tiefling warlock— the twist being that they are actually a several-hundred-year-old elf who is responsible for putting a curse on the land, etcetera etcetera. They picked up an Eldritch invocation that allows them to cast disguise self at will without expending a spell slot, and every hour they recast it. Their eventual reveal is meant to be a huge plot point pertaining to the lore of the world the campaign is set in, so I need to make sure it’s not immediately obvious that they are disguised. How should I go about dming that?

I've had some changeling in my campaign and all I do is not draw attention to the things they do that would reveal this. It has been my players job to not make it obvious. Sometimes I may need to step in and say a few things, if they make an out of character mistake which would reveal this, to try and cover it up. Treat it like you would a secret an NPC would have from the players. If you start over thinking it, it can get stressful and then you could make a mistake. 

I recommend putting bits of lore about this curse and the elf through out. If you want your players to start figuring things out then you want to put them in situations where the tiefling needs to use their elf knowledge, which may not match up with their teifling knowledge. 

That seems dangerous. One errant dispel magic or antimagic cone and the secret is out. Even something as mundane as walking in the rain is risking a smart observer wondering "why is that persons clothing not getting wet?"

So, a great idea. I'm always in favour of PCs living dangerously. :-)

By the book, what was said above. Don't draw attention to it, if a PC thinks something is off let them roll an investigation or roleplay in the group (I tend to avoid PC v PC checks when I can, but it's frustrating to think something is up and not be allowed to look into it). Trust your players want to have fun.

As a dm you could grant the char a limited version of the changeling ability so they can swap between their natural form and their 1 form they present to the party without needing to recast every hour so the usual disguise self tells aren't there (make this special ability dependant on the invocation so you aren't giving the player a major freebie if/when they change invocations) that way tells for the party are roleplay based not blown by being locked together in a room for 1 hour.

any form other than the two above require normal disguise self rules (recast, tells related to actual body shape not matching visual body shape, standard investigate to see through the illusion, ect.)

Casting is too obvious to be hidden every hour of every day. The best bet is to be upfront about maintaining a disguise, but rather for some minor reason like a burn scar or they don't like their freckles or something.

If it were me, I would not put this responsibility on the PC who is limited by PC rules. I'd rather do something like when they cursed the land the curse altered their body, or they have a magic item you made that maintains their appearance. Disguise Self as written is just not reliable enough to cover up something of this magnitude.

I’d watch more than the physical disguise. This character is going to need to be making lots of deception checks as well to maintain their lie. If someone asks them about who their parents were, for example, or lots of other seemingly mundane topics, it will either be a check or a very carefully worded answer, if the player is up to that. I’d make sure to know the rest of the party’s passive insight scores.

Another thing to be careful of is making a single character into a “huge plot point.” What if they die? What if the player moves away or quits? Does that completely derail the campaign? Just be aware it’s really risky.

I'd probably go with a permanent magic item type effect. I don't think it is feasible for a character to cast a spell every hour 24 hours a day, every day of the year, indefinitely and not be noticed. Even elves need to take a 4 hour trance every day and will spend at least 3 hours of that without their disguise unless they wake every hour to recast it.

Perhaps a Ring of Appearances - when worn and attuned this magical ring has the effect of an indefinite Alter Self spell. At the end of a long rest, the wearer can change their appearance within the limits of the Alter Self spell. The ring then maintains this appearance as long as it is worn and attuned. Removing the ring or losing attunement results in the creature reverting to their natural form.

This requires the character to have a "special" ring as part of their backstory. Eventually folks may find out it is magical and the character can describe its effect however they wish (they could even pretend that they don't know what it does). Alternatively the ring could have had Nystul's Magical Aura cast on it so that it no longer radiates magic.

There are too many problems with Disguise Self in the long term. Not the least is that the character's actual skin and body do not conform to the image. Anyone touching them stands a good chance of noticing something is wrong, never mind the effects of wind, rain, snow, weapons, being hit by a spell. and any number of other interactions that should show some effect on a real person but would show no effect on an illusion.

+1 to magic items.

Though it might be fun to cover up an occasional (daily?) recast with nervous bowels and a shy nature.

That’s a very good point the player and I talked over— we made sure their character’s past was the important bit, not their present decisions. But I love the idea of the inconsistent “backstory” being a tell, and I’ll make sure to gather everybody’s passive insight. :)