The dork disc is a term I'd never heard until recently. Normally, it's referred to as a spoke protector or freewheel guard. Essentially it is the plastic or metal disc that sits between your cassette or freewheel and the spokes on the drive side of your rear wheel. The function of this disc is to protect your spokes; should the chain slip off the largest sprocket and into the wheel. The general view is that these discs are normally found on cheaper lower end bikes, and having one on your bike is not cool. The reasons for this seem to fall into two categories, performance and looks.
The performance aspect is normally cited due to weight. It is fair to say that wheels are always a great place to start when trying to reduce the weight of a bicycle, if you can reduce the rotational weight, bringing the bike up to speed will be easier. Even the tiniest additional weight will keep the hardcore weight weenies awake at night.
The look aspect is more of an image thing, some riders will tell you it looks ugly and ruins the look of the wheel. More likely, they will say they dislike the dork disc because of mechanical snobbery. Most chain drops at the rear happen due to an incorrectly adjusted rear derailleur, more specifically, incorrectly adjusted limit screws. If the low gear (L) limit screw on the rear derailleur is correctly adjusted, the derailleur shouldn't be able to move in far enough to push the chain off the largest sprocket and into the wheel. This is the reason given by a lot of riders that object to them, in their eyes having a dork disc is like wearing a sign saying 'I don't do my own maintenance - and if I do I'm crap at it'.
However, the reasons for hating on the dork disc are as much cultural as they are practical. In terms of the looks, on modern bikes the dork disc is generally a clear piece of plastic not much larger than the largest rear sprocket. Yes you can see it, but would you notice it if you weren't looking for it? I doubt it.
Going back to the 70's and 80's the dork discs were normally metal. On very old bikes you will see shiny chromed steel ones, on 1980's bikes they tended to be polished alloy or have an alloy centre with a plastic surround. When clean, they were genuinely attractive additions to the bike. Occasionally I salvage them in good condition from an otherwise ruined wheel and quite a few have been sold, so there must be plenty of riders out there that like them. Obviously beauty is in the eye of the beholder, but surely most people would agree these have a certain charm?
In terms of the performance then, well the pros don't use them. Which could well be the real reason why riders turn their noses up at them, if it is then fair enough - each to their own and all that. But the weight gain is minimal. Even a large chromed steel dork disc will weigh in at less than 100g. An alloy one is about 45g and a small plastic one will be even less than that. Is 25g really going to get you dropped on a club ride? Probably not, but broken spokes might!
A lot of vintage wheels pass through my hands each year, some are refurbished and sold on, some are beyond repair and have to be dismantled. Sometimes rims and hubs can be salvaged, refurbished and sold on ebay or on this site here. Anyway, the one thing I can say, is that when it comes to rear wheels, I see drive side spokes which have the same pattern of damage on a regular basis. Normally it's burring on the spokes down near the elbow, which isn't particularly noticeable at first, but it will significantly shorten the life of the spoke and must be replaced before the wheel is used. This damage is almost always caused by a dropped chain and on occasions could have been avoided with a protective disc in place. It's not guaranteed that a disc will save your spokes, but isn't it worth keeping on just in case?
The argument that a correctly adjusted derailleur won't ever drop a chain is also a bit of a myth, a chain can get knocked into a wheel following a crash. Plus don't forget limit screws can break, get cross threaded, or even work loose and fall out never to be seen again.
Some say that the dork disc actually has aero gains. In reality it probably makes no difference. But the spurious claim is along the lines that disc wheels and deep section rims are considered the fastest, so covering a small part of the spokes with a disc may give a little advantage!
My main pro-dork disc argument is that anything and everything can and does break. To keep our bikes as safe as possible, we should try to diligently reduce the risk of failure. If that means having a part on my bike that may not aesthetically please some, but could prevent me having a long walk home, I'm happy to accept the sneers of other riders.