Is it possible to sharpen a peeler? And if it is, how should I do it?
Are there any things to look out for? (Such as ending up with concave blades.)
This is the sort of peeler I'm talking about:
Sort of. There's a trick to fix up peeler edges: rub the back side of a paring knife along the blade, at roughly the same angle as the peeler blade's edge. Use the tip of the knife if needed. You may need to do this with both sides of the peeler blade.
I suspect the result is closer to honing the edge, but the trick works well enough that I haven't bought another peeler since my kitchen manager showed it to me.
Sypderco ceramic sharpening rods; that is the easiest for me. Very easy to do, holding the rod like a carrot to be peeled, then run the peeler down the edge of the rod. To see how to do it at the end of their three-part training video (their site or youtube). You can buy their sharpening system, or just buy the replacement rods (other companies sell them too).
Or you can use a Dremel, ceramic round tip, either in a Dremel (rotary tool) device or a power drill. Run the ceramic tip along both cutting edges of the peeler (remove burrs of the back side of the blades if you wish). I also use the Dremel technique to resharpen my sewing seam rippers.