Your music demo is supposed to be just that - a demonstration of your music. Demos are used to attract attention from record labels, promoters, managers, agents and other industry types, and what all of these people listen to a demo to find out is what kind of music you make, what unique ideas you have and whether or not you have a sound that it is identifiable as your own.
For this reason, a cover song is not your best option for a demo. Using a cover song short changes you because it stops you from showing off all of your abilities. Remember that your time to capture someone's attention with a demo is very short. If there was ever a time to come with your A game, it's this.
Of course, there are exceptions to every rule. If you're using your demo to show off your vocal abilities or playing abilities - say to audition for a role as a session musician or back-up singer - then a cover song is perfectly acceptable. Further, if there is a cover song that is particularly associated with you/your group - for instance, your fans always shout for it at your shows - then you have a better argument for including a cover on your demo.
At the end of the day, including a cover song on your demo is not an absolute no-no. If you're planning to include a cover on your album, or as previously stated, there is already a cover song associated with you, then you can get away with it. As a rule, however, limit yourself to one, and don't ever just ape the original version. You'll enter karaoke territory, and that WILL be the kiss of death for your demo.
To learn more about making the perfect demo - and what to do with it once you have it - check out this free demo eCourse.