The right thing to do is to ask the copyright owner for permission. In general it's not a criminal offence - you won't go to jail - but you could be breaking the law. It's impossible to say because the law is different in each country.
In general, using the music in your mission if you or your friends play it in the comfort of your own home, is fine. It only becomes a possible problem if you publish the mission on the internet.
Many people don't ask, of course. If you think its morally acceptable, and are willing to take the legal risk, then do it. If not, don't. It's up to you.
Given that you are not permitted (by virtue of your EULA with BIS) to make any financial gain from your mission, some people think its ok if they have obtained the music legally in the first place. (I.e. if they have bought the CD or download.) In this case there is no direct financial loss to the copyright owner.
If you publish a mission with music then it is very easy for somebody to extract that music from the mission and use it for their own purposes. (Use Amalfi's pbo decryptor 1.5 from the Editors Depot.)