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.

Shiitakes are not poisonous and are fine eaten cooked, uncooked, cold, or as leftovers.

Poison's mushrooms contain toxins which cause their effects. Some of these toxins will break down when cooked, but there is nothing I can find about the opposite happening (toxins activating when cooked)

Source: https://en.wikipedia.org/wiki/Mushroom_poisoning

I guess that you heard about the advice not to reheat mushrooms of any kind and thus, to toss the dish if it is not eaten right after cooking (at least I read this advice in a book along with the advice not to reheat spinach and this advice is/was quite popular in Germany). This statement is true if the food will be stored at a place which is not sufficiently cold (<4°C) or not sufficiently hot (>60°C). Mushrooms basically consist of proteins and water and bahaves food-safety-wise like any other (lightly seasoned*) cooked protein like cooked meat: They spoil pretty fast. If you can cool the mushrooms within 2 hours to 4°C or below and store them at this temperature, they will be fine.

Another bad advice is that spoiled food will be fine aber being reheated. It is not. If the food is spoiled, eating it neither hot nor cold is okay. In the opposite case, if the food is safe to eat, it is (food-safety-wise) okay to eat it both hot and cold.