Technically, you need to pay royalties every time you profit off the songs. However, bands don't tend to ever suing a tribute band so long as it's very apparent that they are a tribute band and not the real deal. There's a few artists (I can't remember who - maybe Bruce Springsteen?) that have tried to stop cover bands from covering their songs, but it hasn't been successful, as far as I've heard.
The same is true with any cover band or even a band that does a single cover during their otherwise all original set - they're technically supposed to pay royalties any time a song is covered. However, very few bands care. The few bands that do care probably find very quickly that the legal costs for suing for royalties recovery from playing live are so minimal and the effort to get that money is far greater then it's worth.
Also, claiming that the original band's profits had been negatively affected by the cover band's activities are very, very difficult to prove.
If you're selling records or mp3s of a cover that your band did, again, it's unlikely that a band or the copyright owner will ever sue. However, once your band gets steam, and you're selling copies of that cover song in the 10s of thousands, then you'd better work out a deal quick.
If your cover band takes off, it's probably best to look into it and contact the copyright owners of the song, as well as whomever owns a trademark for a band, and see what they say. Unless you're really making a whole lot of money at it and it's fairly easy to show that you're taking money away from the original band, they're not likely to sue or care.
In essence, you'd be technically breaking the law. Unless you're making pretty good money as a cover band, or if you try to pass off your band as the creator of the songs or as the original band, then you're not likely to have any problems because of high legal costs and government bureaucracy. Whoopie.