We have observed M31 using the ROSAT HRI. We have searched for X-ray emission from supernova remnants (SNRs) previously identified from surveys using narrowband optical observations. We find that a surprisingly small number of the identified SNRs are detected in X-rays at a threshold of ~1035 ergs s-1. The absence of detected X-ray flux from many of the optically identified SNRs suggests that the local ISM density in the vicinity of these SNRs is typically quite low, less than 0.1 cm-3. The few that have been detected likely represent the upper end of the density distribution, having implied ambient densities in the range 0.1 to >10 cm-3. Since H I observations show that all SNRs are in regions with large column densities, which implies densities of 1-10 cm-3, a multicomponent ISM must be invoked. Our measured densities are upper limits for the dominant low-density component of the ISM.