Remove Curse is needed every time
It's right there in the text. "Once you don this armor, you can't doff it until targeted by the remove curse spell or similar magic."
Each time you don it, the restriction begins applying, until lifted by the specified mechanism.
If Remove Curse (or similar magic) said anything specific to indicate an ongoing de-cursifying effect, that would change things, but Remove Curse is in fact quite specific that it doesn't have an ongoing effect on magic items:
If the object is a cursed magic item, its curse remains, but the spell breaks its owner’s attunement to the object so it can be removed or discarded.
Note that all the magical benefits of Demon Armor require attunement, and arguably, they simply misstated the rule in terms of donning rather than attunement + donning, so a DM could rule that donning (without attuning) Demon Armor doesn't require a Remove Curse to doff it (Remove Curse itself is phrased in terms of breaking attunement to cursed items, so it makes sense to think of it as breaking the attunement to allow doffing). But rules as written, any time you don the armor, attuned or not, you must be the subject of a Remove Curse (or similar) to doff it.
Side-note: Interestingly, the wording of Demon Armor differs from Remove Curse in a crucial way: Doffing requires a Remove Curse (or similar) targeting you (the wearer), while breaking attunement to a cursed item via the normal effects of Remove Curse requires the spell to target the cursed object. So, RAW, it's possible to keep the armor attuned indefinitely, even as you don and doff it (at the cost of a Remove Curse per doffing), without needing to reattune (which, if Remove Curse ended the attunement as well, would add a short rest at the beginning of each morning just to regain the magical benefits of the armor after doffing it overnight). This doesn't buy you much (all benefits rely on both being attuned and wearing the armor), but it does mean you can avoid the penalties for sleeping in medium/heavy armor outlined in Xanathar's.
Additionally, there's no strict time limit for doffing the armor after being targeted by Remove Curse (aside from the implicit need for the armor to be donned first), so whenever the party caster(s) have extra spell slots near the end of a long rest (before adventuring begins, and on any day they don't exhaust their 3rd and higher level slots), you could rest, don the armor, have them cast Remove Curse with any leftover spell slots immediately, before slots are restored at the end of the long rest, and you're now able to freely doff it (once) whenever you like. Small reduction in costs of the armor, for small spellcasting cost.