Most forms of healing will not work on undead
The primary drawback to playing an undead is that you are unaffected by the spells Cure Wounds, Heal, Healing Word, Mass Cure Wounds, Mass Heal, Mass Healing Word, Power Word Heal, Prayer of Healing and Spare the Dying.
You are also unable to benefit from a paladin's Lay on Hands or a life domain cleric's Preserve Life ability.
Inflict spells no longer heal undead like they did in earlier editions of the game.
Undead type has no inherent mechanical effect in D&D 5e
However, other than specific spell, effects and items which affect undead different, merely being undead has no inherent effects of its own (D&D 5e Monster Manual, page 6):
Certain spells, magic items, class features, and other effects in the game interact in special ways with creatures of a particular type. For example, an arrow of dragon slaying deals extra damage not only to dragons but also other creatures of the dragon type, such as dragon turtles and wyverns.
The game includes the following monster types, which have no rules of their own.
This is different to D&D 3.5, where monster types had their own inherent attributes, such as an Undead having no Constitution score. Part of D&D 5e's design philosophy is to avoid "hidden" rules like these.
As a result, other than vulnerability to those rare few magic items and effects which specifically target undead, the only differences for an undead PC are those the DM decides. Does your undead have additional abilities like poison resistance? Do they draw the ire of undead-hating crusaders or unsympathetic villagefolk? Do they decay? All of these are up to the DM to choose.
While it's widely believed that undead in D&D 5th edition do not need to breathe, eat or sleep, this is not a game rule, but rather a trait which is common among creatures of the undead type. For example, the Dracolich template (Monster Manual p.83):
The dracolich's type changes from dragon to undead, and it no longer requires air, food, drink or sleep.
Most undead appear to have this ability, in addition to resistance to necrotic damage, immunity to poison, and immunity to being charmed, frightened, paralyzed or poisoned.
However, not all undead have these abilities—they are features of that particular monster, not the undead type. For example, the vampire and vampire spawn do not require air, but are given no special exemption from the other needs to eat, drink and sleep.
As presumably a custom type of undead, it is up to the DM in this case what traits this character gains.
Other game rules which affect undead
The drawbacks of being undead include:
- Cure Wounds, Heal, Healing Word, Mass Cure Wounds, Mass Heal, Mass Healing Word, Power Word Heal, Prayer of Healing, Spare the Dying do not affect you.
- You are subject to a cleric's turn undead, or certain paladins with the same ability
- A ranger with undead as a favored enemy is effective against you
- You are vulnerable to a paladin's Divine Sense and Divine Smite
- You can't benefit from a Life domain cleric's Channel Divinity
- You have disadvantage on an Oath of Vengeance paladin's Abjurt Enemy effect
- You can't benefit from a paladin's Lay on Hands
- 20th level paladins with the Oath of Devotion have advantage on spells cast by you
- 14th level necromancers can command you
- You cannot benefit from antitoxin
- Holy water deals damage to you
- Chill Touch, Sunbeam and Sunburst are more harmful to you
- Commune with Nature and Detect Evil and Good can sense you
- Dispel Evil and Good, Holy Aura and Protection from Evil and Good can affect you
- Forbiddance, Hallow and Magic Circle can exclude you
- You may be vulnerable to an Oathbreaker paladin's Control Undead
- You are vulnerable to the effects of the Helm of Brilliance, Holy Avenger, Mace of Disruption, a Scroll of Protection against undead, Sun Blade, Moonblade, Book of Exalted Deeds, Sword of Kas,
- You can trigger an enemy's Scarab of Protection
- You are vulnerable to a sprite's Heart Sight ability
Some of the advantages include:
- You are immune to the effects of the spells Antilife Shell, Blight, Command, Hold Monster, Phantasmal Force, and Sleep
- A necromancer can't benefit from Grim Harvest for killing you
- You can benefit from an Oathbreaker paladin's Aura of Hate
- You have an advantage when standing on desecrated ground
- You are immune to some of the special effects of the Nine Lives Stealer sword, Sword of Life Stealin and Blackrazor (which now actually heals you if you're hit with it)
- You are immune to various monster attacks, including the Banshee's Horrifying Visage and Wail, the Beholder's Sleep Ray, the Bearded Devil's Glaive effect, the Horned Devil's Tail effect, a Ghost's Horrifying Visage, a Ghast or Ghoul's Claws, a Jackalwere's Sleep Gaze, a Mummy Lord's Blasphemous Word, a Myconid's Rapport Spores, to name a few.
- You benefit from a Death Knight's Marshal Undead ability and a Mummy Lord's lair abilities.