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The Tiefling has the Infernal Legacy racial trait, which grants them the Thaumaturgy cantrip.

In the PHB, it says (emphasis mine):

[...] when you gain a level in this class, you can choose one of the warlock
spells you know and replace it with another spell from the warlock
spell list, which also must be of a level for which you have spell
slots.

Suppose I have a Tiefling Warlock that has just reached level 2. Can I replace the Tiefling's Thaumaturgy cantrip (from their racial trait) with another Warlock cantrip? Or can I only replace specifically Warlock spells of level 1 and higher?

No, for two different reasons.

Most classes cannot replace cantrips at all. Even warlocks and sorcerers can only replace spells of a level for which they have spell slots, which excludes Cantrips since you don't have any 0th level slots.

But even if you could replace cantrips, as the Unearthed Arcana Artificer can, you still can't replace racial cantrips, because your class ability doesn't allow for it.

The warlock class feature says, as you quoted:

choose one of the warlock spells you know

Cantrips you gained from your race aren't warlock spells.

Similarly, the Artificer's ability says:

you can replace one of the artificer cantrips you know

Again, racial spells aren't artificer spells.

First of all, most classes can't replace cantrips, and if they did they would specifically say so.

However, I think that you are getting this a bit confused. I don't think the question here should be whether you can replace cantrips, but whether you can replace that specific cantrip. Since the cantrip is granted to you by your racial trait, it is not considered a warlock cantrip you can not replace or change it without either the DM's approval, or the DM telling you to. It is not specifically said that this is true, but if you look at the official description for the ability (see PHB pg. 43):

Infernal Legacy. You know the thaumaturgy cantrip. Once you reach 3rd level, you can cast the hellish rebuke spell once per day as a 2nd-levei spell. Once you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells

It never says that the spell(s) belong to any given class, even if they normally do. Add to the fact that the spells that you learn later with this ability have restrictions on their usage, it would appear that these spells cannot be replaced (unless the DM says so or makes you) without changing your race (which would also require the DM's approval). Hope this cleared that up!