Linearly polarized Balmer line emissions from supernova remnant shocks are studied taking into account the energy loss of the shock owing to the production of nonthermal particles. The polarization degree depends on the downstream temperature and the velocity difference between upstream and downstream regions. The former is derived once the line width of the broad component of the H$\alpha$ emission is observed. Then, the observation of the polarization degree tells us latter. At the same time, the estimated value of the velocity difference independently predicts apparent downstream temperature on the assumption that the shock is adiabatic. If the actually observed downstream temperature is lower than the apparent one, there is a missing thermal energy which is consumed for particle acceleration. It is shown that a larger energy loss rate leads to more highly polarized H$\alpha$ emission. In comparison of actual observations, our result implies a sizable energy loss rate for the SN 1006 shock.