This is by far the easiest repair to make to a finish. You have two general options, though it's best to do both.
The first option is a coat of wax. Regular paste wax can do wonders for a finish, and properly applied it will give the finish a small amount of additional protection. If you use a dark paste wax on dark furniture then it will help to hide small scratches and whatnot, as well.
The second option, and the one that I prefer as a first step, is to revive the finish by cleaning and oiling it. You can buy commercial "Finish Reviver" liquids, but essentially you can get that effect with mineral spirits and linseed oil.
Take this old drawer, seen below. I was given an old dresser by a lady I worked with. It's a beautiful old thing. Hand-cut dovetails on the drawers, and good, thick burl veneer on the solid faces. It's good, honest furniture. It also looks like it hasn't gotten any love for a long, long time. I've had it for over a year and I know I haven't given it any. Sure, it does a fine job of keeping my underwear off the floor, but it's dull and un-distinguished looking.
Now, look again at that drawer. That washed out look comes from exposure to UV rays (sunlight). A dead giveaway is that the finish is darker and more 'lively' where it's in the lee of the drawer pull... the light must have been coming from the right hand side for most of the dresser's life. This dark spot gives me my first inkling of what the drawer should look like.
I'm going to revive the finish on this one drawer face. First I take a bit of mineral spirits and pour it into a container. I rub the finish in an inconspicuous spot. Why? If the finish was french polish (shellac) then the mineral spirits would take the finish right off. That's not what I want! So I check to see if I can rub any brown off of the finish with straight mineral spirits. Nope. It's probably coated with a cellulose lacquer. Okay, so I wipe the drawer face down with mineral spirits and that helps to remove dirt and prepare the surface. Now I add boiled linseed oil to the bowl. I want at least 25% linseed oil, but 50% wouldn't be a problem. You could make yourself a container of 'reviver' with 25% linseed oil and 75% mineral spirits. I should do that, but I haven't bothered yet. Now I wipe the mixture onto the drawer front. I leave it pretty thick because I'm going to wipe off the excess after 10 minutes, and I'll give it a good buff with a clean cloth. After it dries over night it's good to go.
I put the drawer back into the dresser so you can play 'spot the drawer'. Can you guess which one has the 'revived finish'?
That drawer is looking pretty sweet. Now I have to do the rest of them, 'cause it looks funny.
I sincerely hope the lady who gave me this dresser doesn't read this blog, otherwise she'll be on my doorstep with a shotgun trying to get it back!
Since it's my dresser so it doesn't have to go back to some anxious relative, I'll give the oil a whole week to really dry, and then I'll wax the piece using a dark wax. It's a 'belt and suspenders' thing. The waxing can be done with the drawers in place (and still full of clothes) but you'd be best off emptying the drawers to do the oiling.
This whole job took about 15 minutes to do, and that included waiting 10 minutes for the oil to 'set' before I wiped it off. Doing the whole dresser would probably take 20 minutes. It would be best to stand the dresser on a cloth - if you drip oil on your floor you'll regret it, though it can be cleaned up easily (before it dries) with mineral spirits.
Don't oil the insides of drawers or furniture - it takes forever to dry and it will smell funny until it does.
I'll put another post up about waxing in particular.