Why do my breasts ache when I sleep with no bra?

Lack of support. Possibly due to lack of support of the breast ligaments and tissue pulling on the chest wall muscles. Larger breasts may have more of a problems. Possibly due to not wearing a good supportive bra during the day. See someone about getting evaluated and fitted with the appropriate bra.
Support. Depending on your sleeping posture and size of your breast, lack of support (no bra) for extended time may pull the ligaments and cause pain. Try a sports bra.