If we have a live DJ at our party, is the music licensed?

The answer depends on the DJ and on the source of the music and the location of the event. It may be that the venue is licensed through performing rights organizations (PRO) to have music performances. That would usually include such things as radio, live bands, live DJs, and other sources of music performances in the venue. The venue or DJ should be submitting a cue sheet or list of songs to the PRO after each event. Whether the DJ has been licensed to do whatever it is that is being done to the music, that is a question you'd need to ask the DJ. Each DJ runs their business differently and some are more legal than others. If you are holding your party at a venue that has not bought music licensing from PROs and if your DJ has also not bothered to get licensed through the PROs and for use of the music through the copyright holders, then the music is most likely not licensed.