No you don't need a license to run a Internet Radio Station however if you wish to stream copyright music you do need a license
Licensing Bodies
There are a variety of organisations and bodies who look after music copyright and royalties depending on your country, for example here are the UK’s licensing bodies:
PRS: The “Performing Rights Society” collects royalties on behalf of artists, composers, and cover music that’s played on TV, radio, and online.
PPL: The PPL represents the interests of record labels and collects royalties from radio stations on their behalf.
Choosing a licensing body solely rests on where you intend to broadcast audio, for instance, if your audience is in the USA then you can use SoundExchange.
The following is required for our UK-based clients (as described above):
For online broadcast, two licences are required. One from PPL & one from PRS For Music – for both licences combined, it will cost you just over £300 for the bare minimum package.
We do not force any of our users to purchase these licences and will not suspend any services if we find that they are not licenced. However, if we receive a complaint from PPL, PRS or a music artist/record label – we will be forced to remove the audio file which has been complained about. This could also lead to termination of the main account. You will be notified of this before anything is deleted from your account
IMPORTANT WE TAKE NO RESPONSIBILITY WHAT SO EVER IF YOU DONT HAVE A LICENCE TO BROADCAST ITS YOUR OWN SOLE RESPONSIBILITY TO OBTAIN A LICENCE TO BROADCAST NO MATTER IF YOU OWN OR RENT THE RADIO STATION IN QUESTION