It is forbidden on Shabat to move a candlestick while the candle is burning. There is a halachic principle of “Migo De’itkasa’i” which establishes that something which is muktzeh during Ben Ha’shemashot (the period after sundown) when Shabat begins, it remains muktzeh the remainder of Shabat. Therefore, if a candle was lit in a candlestick before Shabat, the candlestick may not be moved at any point throughout Shabat.
There is a way to make it permissible after the candle is extinguished. Namely, one can stipulate before Shabat that he wishes to move the candlestick during Shabat. By making this stipulation, one avoids the principle of “Migo De’itkasa’i” and is then allowed to move the candlestick after the flame goes out. Although the practice among Ashkenazim is not to rely on a stipulation, the Sefardik minhag allows making this stipulation in accordance with the ruling of the Chida (a”h), in his Sefer Birkat Yosef. The Chida quotes his grandfather, Rav Avraham Azulai (a”h), as commenting that it suffices to make this stipulation just once a year and this allows one to move the candlesticks every Shabat during that year.
Ha’Rav Ovadyah Yosef (a”h) in his Sefer, Chazon Ovadyah writes that once the flame goes out, one who had made the stipulation is then allowed to move the candlesticks for any purpose – whether he needs the space or he wants to protect them from getting ruined. He explains that a candlestick does not fall under the category of “keli she’melachto le’isur – utensils intended mainly for a purpose that is forbidden on Shabat (such as a hammer), which one may not move on Shabat for the purpose of protecting them.
A candlestick is not used for a forbidden purpose, but rather serves as a “basis” (base or holder) for a candle. As such, although it is muktzeh on Shabat, it is not considered a “keli she’melachto le’isur.” Therefore, once a person makes a stipulation, thereby avoiding the prohibition of muktzeh, he is allowed to move the candlestick for any purpose, even to protect it.
For the same reason, Ha’Rav Ovadyah adds that the candlesticks may be moved even if they are made from a valuable silver. The halachah of “muktzeh me’chamat chisaron kis” forbids moving a valuable object which is generally kept in a particular place. However, Ha’Rav Ovadyah states, that only something which qualifies as a “keli she’melachto le’isur” can be considered “muktzeh me’chamat chisaron kis.”
An expensive painting, for example, is not used for any prohibited purpose, as it just hangs on a wall, and it may therefore be moved on Shabat despite its being valuable. By the same token, silver candlesticks are not utensils which are used, but rather hold candles. Hence, they do not fall under the category of “muktzeh me’chamat chisaron kis,” and therefore they may be moved on Shabat after the candles are extinguished.