I only included moves, Abilities, and form changes/Mega Evolutions in this answer. Evolutions don't count, in my mind, but if you want me to include them just tell me and I'll try and find a list somewhere. Also, please tell me if I missed something!
Forest's Curse: Adds Grass-type to the target.
Trick-or-Treat: Adds Ghost-type to the target.
Soak: The target loses their typing and gains the Water-type.
Burn Up: Removes Fire-type from the user.
Roost: Removes Flying-type from the user for one turn.
Conversion: Changes the user's type to the type of the move in its first moveslot.
Conversion2: Changes the user's type to a type that resists or is immune to the target's previous attack.
Colour Change: User loses previous type and gains the type of the move used by the opponent.
Camouflage: User loses previous type and gains the type of the environment.
Protean: User loses previous type and gains the type of the move it's about to use.
Mega Charizard X loses the Flying-type and gains the Dragon-type.
Mega Pinsir gains the Flying-type.
Mega Gyarados loses the Flying-type and gains the Dark-type.
Mega Mewtwo X gains the Fighting-type.
Mega Aggron loses the Rock-type.
Mega Sceptile gains the Dragon-type.
Mega Altaria loses the Flying-type and gains the Fairy-type.
Pimal Groudon gains the Fire-type.
Mega Lopunny gains the Fighting-type.
Mega Audino gains the Fairy-type.
Castform loses the Normal-type and gains the Fire-type in Harsh Sunlight, the Water-type in Rain, or the Ice-type in Hail.
Rotom loses the Ghost-type and gains the Water-type in Wash form, the Fire-type in Heat form, the Flying type in Fan form, the Ice-type in Frost form and the Grass-type in Mow form.
Shaymin gains the Flying-type in Sky Forme.
Arceus loses the Normal-type and gains the type of the plate it holds.
Darmanitan gains the Psychic-type in Zen Mode.
Meloetta loses the Psychic-type and gains the Fighting-type in Pirouette form, when it uses Relic Song.
Hoopa loses the Ghost-type and gains the Dark-type in Unbound form.
Oricorio loses its primary type and gains the Fire-type, Electric-type, Ghost-type or Psychic-type depending on what nectar is used.
Silvally loses the Normal-type and gains the type of the Memory disc that it holds.