All good advise given here . Consider the finish thats on the chairs first ... sometimes in an effort to repair the scratches you wind up doing more damage and its REALLY noticable .
I have used the steam method on gun stocks and it works good , but remember you can damage the surrounding finish if you try it . A wet cloth and your wife's iron ( dont tell her what you are going to do though ;-) ) sometimes can raise the damaged wood , but I'm assuming it just scratched the finish not damaged the wood fibers . Be very careful .
Old English if its a dark wood works good . Or any of the other products mentioned .
If you live around a refinisher , I would let him take a look at it before I did anything . Sometimes you're better off just living with it .