Can I heat and then cool Shiitake mushrooms, and eat cold one day later?
Shiitake mushrooms are certainly not poisonous, either raw, cooked or as you posit in your question, after being cooked, stored in the refrigerator and consumed later. 

If you enjoy eating them cold, go for it. Just keep in mind that like any other cooked food, they can spoil if not stored properly (I have some leftover Chinese mustard greens stir-fried with shiitakes in my fridge right now, and have been eating them for days).

However:

There is a rare allergic reaction in some people called Shiitake Dermatitis which is brought on by eating uncooked or under-cooked Shiitakes. Note that this does not mean it is poisonous, it is a food allergy for all purposes like any other (think shellfish, wheat, soy etc.)

In addition to that, 

If you look, you will probably find advice to cook all mushrooms before eating (this probably has more to do with digestibility than anything though).

Although I believe this is really aimed more at wild harvested mushrooms, in his wild mushroom identification book "All that the rain promises and more", David Arora can be quoted as saying: 


  "With very few exceptions, mushrooms should not be eaten raw. They are
  safer, more digestible and more nutritious when cooked..."


Personally, I would not shy away from eating Shiitakes raw for any safety reason, but you then may be robbing yourself of some of the nutritional value.