It often happens that a person wants to move the Shabbat candlesticks, such as to make room on the table or for some other purpose. Similarly, people sometimes wish to move the Hanukah candles on Friday night after the candles have burned out, out of concern that the children might knock them over, or because they are taking up space. Under what circumstances is it permissible to move candlesticks or a Menorah on Shabbat?
Halacha strictly forbids moving candlesticks on Shabbat while the candles are burning. And even after the candles have burned out, the candlesticks remain forbidden to be moved, as they are considered Mukseh. However, one is able to circumvent this prohibition by making a stipulation at the time of the candle lighting. Before the woman lights the Shabbat candles, she should verbally state that the candlesticks should not be Mukseh and should be allowed to be moved after the candles burn out. This is the explicit ruling of the Shulhan Aruch (Orah Haim 279:4). Hacham Ovadia adds that it suffices to make this condition once a year, stating that for the rest of the year the candlesticks should be allowed to be moved on Shabbat after the flames go out. This is, indeed, a worthwhile custom to adopt in order to enable moving candlesticks on Shabbat without any concern.
There is some debate among the Halachic authorities as to how far this Halacha extends. Several Poskim (including the Mishna Berura and Kaf Ha’haim) maintain that making this stipulation is only effective in rendering the candlesticks a "Keli She’melachto Le’issur." This means that the candlesticks may be moved if they are needed for some permissible purpose, or if the space is needed, but not for the purpose of protecting them (such as if they are in the sun and may become tarnished) or for no purpose. This is also the ruling of Rabbi Moshe Halevi (Israel, 1961-2001), in his work Tefila Le’Moshe. Hacham Ovadia (listen to audio recording for precise citation), however, disagrees, noting that numerous Rishonim (including the Ra’avad, Ramban, Rashba, and Rashbatz), as well as the Hida (Rav Haim Yosef David Azulai, 1724-1806), in his Birkeh Yosef, maintain that the stipulation allows moving the candlesticks for any purpose. And thus by making such a stipulation, the candlesticks are allowed to be moved on Shabbat without any restrictions whatsoever.
Summary: It is forbidden to move candlesticks on Shabbat, even after the candles burn out. However, if one makes a stipulation before Shabbat that he wishes to move the candlesticks after the flames go out, he may, and one may even make such a stipulation once a year to cover the entire year.