Everything has a place. I try to move my tone in the direction of the expected for various cover tunes, but maybe not all the way. Your drummer, unless playing electronic, has only one kit. If the other musicians aren't also trying to tone match from song to song, then the sonic spectrum can get screwed up. If you go to a heavily over-driven tone because the song calls for it, the result may be the bass lost behind the guitar tone.
The bass tone for the various bands you're trying to copy probably doesn't change much from song to song in that band's work. That's because it is what works in the context of the musical style and the sonic space available in concert with the rest of that band. If they had a different guitarist or drummer, they might have a different bass tone.
So, if you're in a band that is going to try to match tone with covers, then go for it. If you're not, though, you have to find a great tone that sits right in the mix. Adding effects or adjusting EQ to more closely match a tune is fine as long as you don't lose the core. That's especially true in the bar band world. My job there is to groove so that people bounce around, have fun, and get thirsty. That's much more important than a perfect rendition of a song, and a meaty core to the tone is one of the keys.
I walk a fine line in a three piece band plus vocals. I like to mix up the tone a bit here and there so all the songs don't sound the same, but you can't go too far afield with no other support.
By the way, I generally plug into a Bass Pod XT and use the modeling features. I don't use heavy effects from the Pod, just a little light chorus on some tones and an external EH POG here and there for octave up effects.
-bob