Is surgery for an anal fistula more painful than just leaving the fistula alone?

No: Typically a fistula is a chronic inflammatory "tunnel" between the anus and the skin. There are many types, many treatments, and the most common surgery involves literally "filetting" open the tract to allow it to heal by itself, and thus obliterate the tunnel. This is somewhat painful, but offers the advantage of removing the chronic, longstanding focus of inflammation and pain, and so is usual.

No: In the short run (the first few days after surgery), yes. However, the surgery will usually cure the fistula and eliminate not only the pain that you may be experiencing before the surgery, but prevent future abscesses (and the need to drain them), not to mention other complications of the fistula. So, in the long run, it will be less painful to have it corrected.