The most common vocal removal techniques work by taking advantage of the fact that the vocal is usually dead center in the mix (it is equal on both left and right channels) whereas most other instruments are mixed off to one side or the other. (Bass is often dead center as well but occupies a lower frequency range so the cancelling algorithm or circuit can keep it from being cancelled).
So basically, songs that are mixed any other way will not cancel as well. Particularly, you may get reverb or delay of the vocal remaining although the basic vocal itself is gone. Background vocals which are mixed in stereo may remain.
I know that there are some fantastic plugins out there these days for isolating parts in already-mixed tracks using more advanced techniques.. but these are not generally available for free in a program like Audacity (great as it is.. I use it myself).
I would also suggest looking for a STEMS format version of your song. STEMS is a multitrack format for DJ's that allows easy manipulation of specific elements.. and the lead vocal would surely be on a track by itself.