We have a 20 year old leaking skylight, and right now I am contemplating either removing or replacing it. Not exactly sure why it is leaking, perhaps the flashing is bad, but it's leaked for a while. One thought was perhaps it had to do with the actual skylight mechanism (this one used to open) So I'm thinking of replacing it with a sealed one or roofing over it. I've read a bit about the latter, and it seams to do it properly it's quite a bit of work, you have to tear out the framing structure, so that air flows to the peak, and make sure it is sealed an insulated like the rest of the structure. On the other hand, replacing the skylight will require re - flashing and perhaps changing the size of the frame a bit.
The real answer is: Don't do anything until you've identified the root cause. Find the leak. FLashing and sealant can be replaced a lot more easily than a whole new unit, not to mention the difference in cost and time.
Similarly, if there's a leak where the glass (and frame) separates from the support frame as you open the skylight, address that specifically.
Never start out by attempting the most expensive solution.
If it's a composite roof, I think it's a little more work to roof over the skylight just because of the interior sheet rock repair, but much more expensive to buy a new skylight. You don't need to do any framing to tear out the skylight, but to do it right you should remove the sheet rock from the skylight walls, and repair the hole in the ceiling. If you replace the skylight, go with the velux (or similar price point brand). The flashing kits that are made to go with them use step flash and are very effective in preventing leaks.
The best way is to install a curb mounted skylight. Almost all new skylights are done this way. A wooden frame, usually from 2x4s, is mounted, sealed and flashed, then the skylight mounted on top of the curb with screws. If there is ever a problem with the skylight it can be replaced in minutes. Most often problems occur with how the skylight was mounted to the roof. Get an installer who guarantees the roof work for life, then buy a skylight quaranteed for life and you'll never have a worry.
Remove for sure. You can replace a skylight, throw down flashing around it, caulk it up and basically do a really good install in about 1-2 hours (not accounting demo as this is all over the place with skylights) per light.
If you totally remove it you are going to have to do a lot of patchwork on the roof and a lot of shingle repair. You might have cut past the hole for your skylight and replace adjacent boards or at least cut them - depends on how skylight was installed. Also you may need to replace rows of shingles below skylight - it can be a mess and costly.
On leaking - THEY ALL LEAK. Their seals go bad sitting in the sun. They require maintenance and possibly replacing every 10-15 years. There are things that are easy to do to make your seals last longer but that is a different question. But I would only get rid of my skylights if I was reshingling the whole roof. As it stands to remove it you are probably reducing the value of your home and costing yourself more money.