Never. Only wear a button-down collar with a sports jacket, and no tie.
This is controversial. Button down collars are not considered appropriate with suits in the UK, but in the US, particularly in New England, they are considered a classic choice.
A button down shirt is a sports shirt and should not be worn with a suit and tie. The button down collar was originally developed for use in Polo matches to keep the collar from flipping up against the rider’s cheeks and face. It came to be used on some other sports shirts as a style, but should never be worn with a tie. Wearing a tie with one simply demonstrates lack of knowledge of proper men’s dress. Of course, it’s each individual’s right to dress as they wish. However, men who actually know something about style, cut of clothing, (such as modern suit jackets being cut in a way that the bottom button should not be done up, but simply used as decoration) will be smiling inside when they see that someone doesn’t know how to properly dress. But don’t worry, we will stay polite and not say anything. 🙂