GMOB, yes, they can. Cancerous tumors to be more exact but most tumor cases are benign and as long as it is not cancerous, the fish can live until it meets its end. Cancerous tumors on the other hand will eventually take over and kill the fish.
Steve, how big is the betta? I was going to suggest dabbing the growth with iodine solution carefully using a Q-tip and see if this might inhibit the growth. Use gloves to handle the fish or apply stresscoat on your hands as rough human skin will only peel off the slimecoating and leave the fish vulnerable to skin issues.