What is your location? As this may have an impact...
I am not directly employed in law so this is just my opinion from reading rather than advising;
I think that you are putting yourself in a risky position by publicly playing tracks without obtaining permission from the copyright owners...
If you are not actually distributing the tracks by allowing others to make copies then I think that you are unlikely (again, in my opinion) to be dragged to court but I still don't think that the owners of the tracks (artists and/or record companies) would approve without you getting permission from them first.
This is something that helped me come to the above conclusions;
Using the work of others
If you use samples of music by other authors in your work, ensure that you get permission to use the work before you attempt to publish or sell your work. Similarly, if you use loops or samples available via sample collections etc.
ensure that these are licensed as free to use, or obtain permission first.
Source:
http://copyrightservice.co.uk/protect/p07_music_copyright
Please note, however, that I am considering your website to be "your work" as mentioned in the paragraph above.
Again, your location may have an effect on the guidelines to use, though I don't assume there will be a large difference...?
Hope that helps!