Could I turn Mistform Dreamer into a Shrine?
No. Alas, Shrine isn't a creature type, it's an enchantment type.
Rules reference:

205.3h Enchantments have their own unique set of subtypes; these subtypes are called enchantment types. The enchantment types are Aura (see rule 303.4), Curse, and Shrine.
205.3m Creatures and tribals share their lists of subtypes; these subtypes are called creature types. The creature types are Advisor, Ally, Angel, Anteater, Antelope, Ape, Archer, Archon, Artificer, Assassin, Assembly-Worker, Atog, Aurochs, Avatar, Badger, Barbarian, Basilisk, Bat, Bear, Beast, Beeble, Berserker, Bird, Blinkmoth, Boar, Bringer, Brushwagg, Camarid, Camel, Caribou, Carrier, Cat, Centaur, Cephalid, Chimera, Citizen, Cleric, Cockatrice, Construct, Coward, Crab, Crocodile, Cyclops, Dauthi, Demon, Deserter, Devil, Djinn, Dragon, Drake, Dreadnought, Drone, Druid, Dryad, Dwarf, Efreet, Elder, Eldrazi, Elemental, Elephant, Elf, Elk, Eye, Faerie, Ferret, Fish, Flagbearer, Fox, Frog, Fungus, Gargoyle, Germ, Giant, Gnome, Goat, Goblin, Golem, Gorgon, Graveborn, Gremlin, Griffin, Hag, Harpy, Hellion, Hippo, Hippogriff, Homarid, Homunculus, Horror, Horse, Hound, Human, Hydra, Hyena, Illusion, Imp, Incarnation, Insect, Jellyfish, Juggernaut, Kavu, Kirin, Kithkin, Knight, Kobold, Kor, Kraken, Lammasu, Leech, Leviathan, Lhurgoyf, Licid, Lizard, Manticore, Masticore, Mercenary, Merfolk, Metathran, Minion, Minotaur, Monger, Mongoose, Monk, Moonfolk, Mutant, Myr, Mystic, Nautilus, Nephilim, Nightmare, Nightstalker, Ninja, Noggle, Nomad, Octopus, Ogre, Ooze, Orb, Orc, Orgg, Ouphe, Ox, Oyster, Pegasus, Pentavite, Pest, Phelddagrif, Phoenix, Pincher, Pirate, Plant, Praetor, Prism, Rabbit, Rat, Rebel, Reflection, Rhino, Rigger, Rogue, Salamander, Samurai, Sand, Saproling, Satyr, Scarecrow, Scorpion, Scout, Serf, Serpent, Shade, Shaman, Shapeshifter, Sheep, Siren, Skeleton, Slith, Sliver, Slug, Snake, Soldier, Soltari, Spawn, Specter, Spellshaper, Sphinx, Spider, Spike, Spirit, Splinter, Sponge, Squid, Squirrel, Starfish, Surrakar, Survivor, Tetravite, Thalakos, Thopter, Thrull, Treefolk, Triskelavite, Troll, Turtle, Unicorn, Vampire, Vedalken, Viashino, Volver, Wall, Warrior, Weird, Werewolf, Whale, Wizard, Wolf, Wolverine, Wombat, Worm, Wraith, Wurm, Yeti, Zombie, and Zubera.