A tabaxi ranger/assassin in my game made a pact with an unknown power (homebrew deity-like undead necromancer from the Death Plane). To achieve this he became an undead tabaxi (and he took one level in warlock). So now he is a creature that's both humanoid and undead.

Party composition (level 10): paladin, eldritch knight/wizard, ranger/assassin/warlock, rogue, warlock.

What are the mechanical consequences, both in terms of benefits and penalties, that arise for a PC with both humanoid and undead creature types?

The most prominent issue is healing.

The majority of healing spells (among them the most frequent ones, Healing Word and Cure Wounds) don't affect the undead. There are still some ways to heal yourself (see this question for ways of healing the undead), but it's "harder" to heal you. Most, if not all, healing spells that rule out undead creatures use the following wording:

This spell has no effect on undead or constructs.

Your paladin's Lay on Hands won't work on you either, for the same reason.

I think it's safe to assume that the mixed-type player character is affected by the restriction.

The best guideline we have on the topic is the Unearthed Arcana article that introduces Centaurs and Minotaurs as playable races. Those two races have the following trait:

Hybrid Nature. You have two creature types: humanoid and monstrosity. You can be affected by a game effect if it works on either of your creature types.

OP's homebrew player race is a mix of humanoid and undead (however that works), but the principle is the same. Note that Unearthed Arcana is only beta-test material, so this is not RAW, and the trait might change if the races are ever released (if they even keep their hybrid nature at all).

Furthermore, the restriction on healing undead specifically rules them out - it doesn't state "this spell works only works on creatures that have a type other than undead or construct". In that case, you could probably affect mixed-type creatures, but the current version is unambiguous in my opinion.

Other issues include clashes with Cleric and Paladin abilities

Most prominently, you would be affected every time a Cleric or Paladin in your vicinity use Channel Divinity for e.g. Turn Undead. That won't be as much of an issue since your party doesn't have a Cleric, but you might still be accompanied by a Cleric NPC, or a player that later joins your group might want to pick one. Also, you do have a Paladin - and while they don't have access to Turn Undead, depending on their subclass, they might have Turn the Unholy, which will affect you as well.

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.

Undead traits

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.

An undead PC becomes immune to effects that omit undead, and becomes a valid target for undead-targeting effects.

This means becoming immune to certain spells and effects, both helpful and harmful. It also means the PC becomes susceptible to additional harmful features that are designed to harm undead. Overall, gaining the undead creature type is not very advantageous.

As in Quadratic Wizard's answer, there are no built-in mechanics to being undead. Many undead are evil, or immune to exhaustion or poison or disease, but some are not. So we cannot assume that becoming undead automatically causes the PC to gain or lose any features. Technically, an undead can still benefit from rest, be charmed or confused, or grow old with age.

There are two major changes. First, the PC would become immune to any effect that specifically omits undead or has no effect on undead. There are pros and cons. Some notable examples include:

• bypassing Antilife Shell
  


• immunity to Blight, Cause Fear, Command, Sleep


• if killed by a necromancy wizard, the wizard won't gain hit points via Grim Harvest


• cannot be healed by Cure Wounds, Heal, Healing Spirit, Healing Word, Mass Cure Wounds, Mass Heal
  


• cannot be healed by a paladin's Lay on Hands feature


• cannot benefit from Raise Dead, Spare the Dying
  

Second, the PC is now a valid target for spells and features that apply to (or have additional effects for) undead; most of these are harmful to undead. Some notable examples include:

• a cleric's Turn Undead and Destroy Undead features


• a necromancy wizard's Command Undead feature


• a paladin's Divine Sense feature, +1d8 damage from Divine Smite, and various other features


• various ranger features if they chose undead as their Favored Enemy


• 
  Detect Evil and Good, Protection from Evil and Good, Dispel Evil and Good
  


• 
  True Resurrection would restore them to their non-undead form

There may be some unusual consequences of a PC having multiple creature types (both humanoid and undead). Many effects in 5E assume that creatures have one type; some spells that target a certain type don't explicitly omit others (e.g. Charm Person, Dominate Person, Calm Emotions, Reincarnate). With two creature types, this is no longer the case. The PC is now susceptible to both humanoid-targeting and undead-targeting abilities.