I can't positively say for fish since I haven't bred any but I show Great Danes in akc conformation so can tell you some about that that I suspect would apply to any animal. You inbreed to strengthen certain desired characteristics but you also increase the chances of undesirable traits at the same time such as deformities. I.e., my present dane, Hudson, has an absolutely to die for gorgeous head but if he had an ancestor along the way with hip dysplasia, or worse yet if HE had hip displasia, and I bred him back to a littermate I increase my chances of getting at least one pup with a gorgeous head but also increase the chances that the pups will inherit hip displasia. It should only be done when you REALLY know your dog's ancestry and after testing for known problems to the breed! (Fortunately Hud's ancestry is clean and he tested clean so he now has 5 adorable new puppies!!!!!!!! http://danemarkdanes.com/litter.html)
Genes are genes so I would think that fish would be no different in this respect.
A few things to consider would be why and to what end? I've been straining genes for a couple years now. Why, because I want show stopper fish. Whats been said about the pro's and cons are petty much spot on, although there was one issue missing. Mentaly, multiple inbreds are inferior. New blood every third generation is a plus.
To what end do I do this? I do this with the intent of adding colorfull, easy to keep, fresh water fish to the hobby some day. Oh, and because I can.
...sorry, but on a serious note I do have an inbred Cyphotalapia frontosa. It has seven stripes on one side and six on the other. Some fish look cool, but in most cases have lower resale/trade in value. However, there is a pic floating around on MFK of a Flowerhorn that had a heart shaped pattern which fetched a pretty penny. If you can do it then try and see what you get. Just dont trade them or sell them to anyone else.
my guppys breed non stop, i started with 1 male and 2 females that where feeders that i threw in my loach tank. now i am on the 5th gen of guppys with about 20 of the little guys swimming around. kinda gross but it works.