I saw the titular question over at Quora and thought I'd offer up an answer. Unless the bylaws so provide, shareholders have no right to remove a CEO. Delaware General Corporation Law sec. 142, for example, provides that: “Officers shall be chosen in such manner and shall hold their offices for such terms as are prescribed by the bylaws or determined by the board of directors ….”
In general, the practice is that the Chairman of the Board of Directors must be a member of the board of directors. In a few states (such as Ohio, if memory serves), this is statutorily required. In most, it is an understood best practice. At its first organizational meeting of a new year, the board will select one of its members to serve as Chairman. The chair serves at the pleasure of the board and can be removed from that position by the board at any time. But there is no provision under Delaware law for the board of directors to remove one of its members. Indeed, there is case law specifically holding that the board lacks power to remove one of its own members. See, e.g., Dillon v. Berg, 326 F. Supp. 1214 (D. Del. 1971). Hence, while the board may deprive the chairman of his position as chair, the removed chair will remain a member of the board.
Conversely, the shareholders may not remove a sitting chair from the chairman’s position they can remove a sitting chairman from the board. Under Delaware General Corporation Law § 141(k) shareholders are authorized to remove a board member. that right cannot be revoked in either the bylaws or articles.