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.
You could theoretically pop the blade out of the handle and try and run it over some sharpening stones (by no means easy with the odd shape), but considering that would take time, effort, and at least a £5 sharpening stone, you're probably better off just dropping £1.50 on a new peeler.