Donuts can be put in the fridge, but it's generally not recommended for the best taste and texture. The reason is that refrigeration can cause the donuts to stale faster and lose their freshness.
When you store donuts in the fridge, the cold temperature can promote the crystallization of the sugars in the donuts. This process can lead to a change in texture, making them harder and less appealing to eat. Additionally, the fats in the donuts may solidify in the refrigerator, resulting in an unpleasant mouthfeel when you eat them.
If you have donuts that you won't consume within a day or two, it's better to freeze them instead of refrigerating. Freezing helps to retain their freshness and prevents staleness. To freeze donuts, place them in an airtight container or seal them in a freezer bag. When you want to eat them, simply let them thaw at room temperature, and you can even warm them up in the microwave or oven for a short time to restore some of their original freshness.
In summary, while it's not ideal to store donuts in the fridge due to potential texture and freshness issues, freezing is a better option for long-term storage if you don't plan to consume them right away.