On my lathe, I don’t think about cleaning the inside of the spindle or tail stock, but I should. Gunk can build up in the morse taper of these areas and throw off how well your drive and tail centers seat, center, and hold your work. The gunk can also cause damage to these tools.
But cleaning the morse taper is difficult. It’s a small area – I cannot use a steel brush as it will eventually damage the morse taper.
However, I saw a tip in American Woodturner that helps a lot — use a #20 shotgun cleaning brush. The brush ($2-$3) itself does not have a handle but I’m a woodturner, I can make one out of a pen blank. That is what I did.