If you are streaming, the stream is public or broadcasting to the public, and you are playing copyrighted material you DO need to license your stream. Or you would need to get permission from every record lable to play their music. It is called getting public performance rights. Honestly its easier to just get a stream license.
Luckily we live in an internet age and getting a license for your stream is really easy and is becoming affordable. StreamLicensing.com is one of those cheap and easy ways to make sure you are legal with your stream.
It is very important that whoever owns the material/music you are streaming to get paid what they are owed. When you purchase a stream license you are ensuring that you are not stealing their work, and they are getting proper credit.
Technically if you are playing music in a public place no matter what form you need to pay for it (although its not really enforced)
Of course this is a lightly enforced unless its a huge venue or something as I dont think they are going to walk around city streets finding every street performer or dance instructor. But I hope you see the point :)
As mentioned earlier there are really really easy ways to get licensed without having to go through all of the big names (ASCAP, BMI, etc).