I'm wondering if removing horns and tail is possible for a Tiefling, and if so, if it causes damage and/or permanent hit point loss.
I am also wondering if said horns and tail grow back in time.
The question also applies to the pointy end of (Half) Elf ears, to the big teeth of a Half-Orc, etc.
No material specifies that they cannot, nor any mechanical effects or penalties for a tiefling that is mutilated in such a fashion. The nearest we get is that the DMG, on page 272, includes an optional rule about lingering injuries which has examples including severed limbs. For instance, losing an arm:
Lose an Arm or a Hand. You can no longer hold anything with two hands, and you can hold only a single object at a time. Magic such as the regenerate spell can restore the lost appendage.
So losing a body part does not inherently incur permanent HP loss. However, another example is the "festering wound", which does reduce your HP maximum over time until the wound is healed magically or otherwise, so a tiefling that is mutilated might suffer such damage due to a lack of appropriate medical care:
Festering Wound. Your hit point maximum is reduced by 1 every 24 hours the wound persists. If your hit point maximum drops to 0, you die. The wound heals if you receive magical healing. Alternatively, someone can tend to the wound and make a DC 15 Wisdom (Medicine) check once every 24 hours. After ten successes, the wound heals.
For a significant injury to the head/face such as that incurred by removing the horns, the "horrible scar" injury might be appropriate:
Horrible Scar. You are disfigured to the extent that the wound can't be easily concealed . You have disadvantage on Charisma (Persuasion) checks and advantage on Charisma (Intimidation) checks. Magical healing of 6th level or higher, such as heal and regenerate, removes the scar.
I would also be inclined to say that a tiefling that is suddenly missing their tail should be at disadvantage on acrobatics and athletics checks for a while at least, since for any creature with a tail that large it will significantly affect their balance, but could get used to it with some time.
In any event, whether or not permanent hit point loss is incurred, the removal of the appendages would be a significant physical trauma that should definitely cause hit point damage at the time, which would need to be healed like any other.
In nature, damaged horns may or may not regrow based on the extent of damage done to them and the type of horn. True horns, such as those possessed by goats and cattle, are live bone with a keratin exterior, and have a blood supply and such. They are not like antlers (which are dead bone, shed and regrown every year) or other horn-like growths (such as that of the rhinoceros, which are pure keratin and will regrow if severed, like a fingernail).
The depictions of tiefling horns in official artwork seem most like true horns in the style of bovine animals. Chips to the exterior of a true horn will probably regrow and limited damage to the bony core could heal, but if snapped or completely removed they won't just grow back on their own (though a snapped horn can be set like a broken bone and may heal back together).
A severed tail is unlikely to regrow much if at all - most creatures that lose tails might regrow a very short stump at best.
In both cases being subject to the Regeneration spell would immediately cause lost appendages to start regrowing:
The target's severed body members (fingers, legs, tails, and so on), if any, are restored after 2 minutes.
As per this question, a character probably cannot pick-and-choose which body parts they would regenerate if affected by the spell, so the regeneration of horns and tail would occur even if the subject found that an undesirable effect.
In the general case, assume that people heal like they do in real life unless otherwise stated. If you pull out a person's teeth they don't grow back, nor if you cut off parts of their ears.
This is going to be a pure rules as interpreted/real life facts check, but here goes.
If the part of your body that is cut off can bleed, it causes damage. If your character doesn't regenerate or heal naturally, anything cut off does not grow back. Horns would be the DM's option, but almost always no, they wont grow back (see below for why) as horns are traditionally drawn based on the designs of existing animals, like goats (short), gemsbok (long), bulls (thick and large), kudu (spiral straight), and rams (spiral curved).
http://www.wildliferanchtexas.com/horns-antlers/
So if your Tiefling or similar horned race actually has Antlers, like the Peyton or Great Stag, then those antlers do grow back. But horns don't grow back.
Exception: if the horns are a magical curse, or the consequence of a magical effect, then the cursed aspects of the form restore themselves traditionally. So if a witch, olympian god from Clash of the Titans, or a spell curses you to look like a monster, making cosmetic attacks on yourself to look more human will not work.
As to whether it causes permanent hit point loss or hit point loss in general? Yes. In livestock dehorning, anesthesia is used with sedatives and a vet is hired to reduce bleeding, cauterization is used etc. In other words, it would be extremely painful and cause damage.
The amount of damage it would cause? Probably, again, this is all house rules, but, a bigger horn causes more damage, so the amount of damage your horns can cause is probably the amount of damage you would suffer losing them (so d4 horns would cause d4 damage, d6 horns would cause d6, etc.) Whether that eats at your permanent hit points is the DM's prerogative.
Either way, if you have Antlers, they grow back, and if you lose them, no, they shouldn't cause permanent hit point loss, because those animals are designed to lose their natural weapons and replace them.
In nature, if an animal with horns loses them, which is only possible by the horn being removed, the horns will regrow. For example, if a rhino is dehorned without cutting into the skull, the horn can regrow to it's original size within three years. This true of other species with horns as well.
The difference between horns and antlers is that antlers are a temporary fixture upon the animals head, falling off at the end of the mating season, and regrowing the next mating season, whereas horns are permanent, and never stop growing.
Furthermore, in most antlered species, only the males will develop them, and only during the mating season. An exception would be reindeer, in which the females grow antlers during the winter months, outside of the mating season, however, male reindeer only have horns during the mating season. Horns are more like hair, whereas antlers exist to allow a species males to fight each other so as to get it on with all the does.
The above matters if you are trying to model a Tiefling based on how horns work in the real world. If that isn't your model, then the rules leave it open to DM discretion.
