It has been said that the worst thing you can drink with cheese fondue is a lot of iced water (a small glass should be all right) as too much of this will cause the cheese to solidify in your stomach and you may have digestive problems. Red wine is not recommended either - the tannins will be too strong with the warm cheese. This is more about taste than health; though if you find the tannins in red wine don't always agree with you, the effect will probably be worse if you drink it with strong, acid cheese.

The best thing to drink is probably a dry white wine or dry cider. White wine is used in the preparation of fondue, so a similar wine to the one used would make a good match. In some traditions, you drink black tea with it, or a small glass of grappa or similar spirits. None of these drinks should cause any digestive problems.