To replace or repair ? My personal feeling as to replacement is...if it's a very old piece that would benefit from remaining original (rare, valuable, irreplaceable, everything else is original) then the extra effort to preserve it should be attempted. But I have come across quite a few 100+ year old boards that were simply falling to pieces, not only did removing the frets cause chipping but the force of installing new ones did as well. Ugh.
If the instrument is a relatively new (30 years old or less), it is often a simple matter to order a new board directly from the factory and go from there.
Removing the Fingerboard
I personally use a special heating blanket made specifically for this purpose and heat the board prior to removal. (In fact at one time I used a good old clothes iron, however, they can slip around a bit and place heat over the edge of the fingerboard.)
Because the neck and fingerboard are assembled before painting, separating the two breaks the finish line which will usually require touch up. Plastic inlays (not pearl or abalone) and binding will at times require replacement as the heat necessary to soften the glue joint between fingerboard and neck can damage vulnerable pieces. This is especially true if the frets have already been removed which places the heat directly on the board, I avoid that if at all possible if not replacing the board.
I normally buy all of my fingerboards pre-slotted from the factory or other suppliers since it's cheaper to buy a slotted fingerboard than to make it from scratch. And obviously there are things better left to machine, this is one of them.

Many of us like to use removal pins when gluing fingerboards in position. After drilling a small hole thru the fret slot (or beneath an inlay if necessary) and into the neck these pins will help to keep the board in proper position while clamping. And believe it or not, that can be quite a challenge, especially when it's not new construction or you're working on a neck that is still glued to the body. Because our tiny holes are made beneath the fret or inlay all evidence is concealed.
Some manufacturers use pins as well and I have come across some sort of locator pins when removing a factory board on occasion.
Fingerboard Inlays - Although I do not offer custom inlay (sorry, don't have enough time to go there) I can replace small pieces of abalone or pearl inlay which may be missing or damaged.
If you are interested in seeing or obtaining some wonderful custom inlay on your instrument I strongly encourage you to visit either Larry Robinson's site or Bryan England's Custom Inlay site.

Shrunken Celluloid Inlay
Replacing Inlays - Here's a simple example....One of the simplest upgrades to inlays may be changing out plain white pearl dot markers for more colorful abalone pieces. And while it's not rocket science it can call upon your patience.
If I must save a piece of pearl inlay I will usually use a clean soldering iron to remove it it. I place the tip of the iron on the inlay to heat the glue which usually likes to bubble out beneath when good and hot. You can tattoo a board with a iron so great care is necessary.
Plastic and celluloid can not be heated and will go poof if touched with an iron. Once the adhesive has let go I try to press on one end of the inlay in hopes that the opposite will elevate slightly above the board and allow me to lift it out. This is not always as easy as I'm making it sound but I try to avoid picking at the edges as you can create a "picked at" look when all is said and done.
Most of the time however, saving an inlay is not necessary and I can simply drill thru it's center and lift it out or pop it out in pieces.
Once the old slot is clean and you've made or chosen a replacement inlay which fits well but is slightly higher than the board it can be glued in. Tinting your adhesive to match the board can greatly improve the end result if there are any gaps between inlay and slot. I personally like to use epoxy or superglue here.
Once the adhesive is cured the inlay is filed level with the board (or bridge or whatever you're inlaying). Here is where a good deal of folks mess up. That pearl inlay is harder than the wood and if you try to use sandpaper or such you will surely end up removing more of the board than the inlay. I like to use small bastard files to file it down to the appropriate height and then I may switch to scraping or sanding.