This is one of those things I've been curious about for years, but I haven't taken the time to set up enough of an experiment to give it a fair shake. Stringing a double bass is a bit of a pain, and doing it twice in order to compare the two would be twice the pain. I also have a very slightly bent D tuner so it's a little stiff to turn, meaning by the time I'm done my arm/hand isn't happy. If you or someone else is bored, it would be great to hear a recording of the strings with the washers and then without on the same bass.
My two cents or nickel because we round and the CAD is horrible right now? It really doesn't make much difference. If there is a sharp end of the core sticking out it might prevent that from poking the tailpiece, but you can usually position the ball in such a way that you don't have that problem. It might protect the back of the tailpiece slightly, but most of them are Ebony or something else pretty hard that you don't have to worry about the ball punching a hole through your tailpiece. If you do, a felt washer isn't going to stop that from happening. Unwanted buzzing/vibrations seem pretty unlikely given how much tension is on the string holding it there. As far as a change in the sound, while the whole string does vibrate and tuning the afterlengths can make a difference, you have both the bridge, and the "nut" on the tailpiece between the sounding part of the string and the ball end, so you really aren't going to get any more/less vibration out of it or have to worry about muffling things.
That's all educated guesses considering I haven't conducted the experiment myself to observe any difference, but I wouldn't lose too much sleep over it.
