How much damage can Trojan viruses do?

My boyfriend has got a Trojan virus on his computer which he got from a torrent. He can't remove it and he doesn't have the money to get it fixed. Should he disconnect it from the Internet or is it safe to carry on using? So far it has changed his screensaver, downloaded some random songs and keeps blocking his Internet. Can anything bad happen if he just leaves the virus and keeps using his laptop?