If a Wraith reduces your maximum HP to zero, you die
The explanation you've been given seems to be a little mixed up - it's not that necrotic damage inherently cannot be healed, but that the Wraith's attack deals necrotic damage and also has a secondary effect which can reduce your HP maximum until your next long rest:
Life Drain. Melee Weapon Attack: +6 to hit, reach 5 ft., one creature. Hit: 21 (4d8 + 3) necrotic damage. The target must succeed on a DC 14 Constitution saving throw or its hit point maximum is reduced by an amount equal to the damage taken. This reduction lasts until the target finishes a long rest. The target dies if this effect reduces its hit point maximum to 0.
In fairness, lots of maximum-HP-reducing effects are associated with something that does necrotic damage, so a bit of mixup is understandable. However, most of the time necrotic damage has no special effect on your hit point maximum and can be healed like any other kind of damage - the particular type it is is usually only relevant for characters and creatures with immunity, resistance, or vulnerability to certain kinds of damage.
Most importantly for your question, you should note the final sentence of the attack's description - if the Wraith's attack would reduce your maximum HP to zero, you just die outright. It takes a bit more than a long rest to recover from that!
As far as I'm aware, all effects which reduce maximum HP in this way clarify that you die when reduced to zero HP max - and possibly something else dire happens, like then turning into an undead creature of some kind. But if you have taken some maximum HP reduction which hasn't killed you, either because it's not quite enough yet or it's specified to do something non-fatal when you reach zero - the spell Greater Restoration is capable of undoing the damage:
You can reduce the target's exhaustion level by one, or end one of the following effects on the target:
- One effect reducing the target's hit point maximum