Ultimately no. Not all mouthwashes contain alcohol, but the ones that due are generally small in proportion to every other chemical Example:
Alcohol in MouthwashListerine 26.9%Scope ...
Alcohol in mouthwash may cause you to fail a breathalyzer test because the residual alcohol remains in your mouth for up to 15 minutes after using. The breathalyzer will pick up this residual alcohol...