Generally speaking, it is forbidden to read by the light of oil candles on Shabbat. Therefore, if the electric lights went out in the dining room on Shabbat, either because of the timer or because of a power failure, and the room is illuminated only by the light of the oil Shabbat candles, it is forbidden to read in the room.
There are, however, several exceptions to this rule. First, if a person knows the text he is reading from memory, and he uses the book only for additional help, then he is allowed to read the text by candlelight. Thus, for example, if one wishes to sing Pizmonim, and he generally knows the words by heart, then he may read the songs from a book by the candlelight, since he does not actually need the light to read. Likewise, if the night of the Seder falls out on Friday night, and the only light in the room is the light of the Shabbat candles, one who for the most part knows the text of the Haggadah from memory may read the text from the Haggadah. Others should read along together with the one who knows the text from memory.
Secondly, it is permissible to recite Birkat Ha’mazon by candlelight on Shabbat. For one thing, many people more or less know the text by heart, and use the book only to enhance their concentration. And even if one does not know the text from memory, the Torah obligation of Birkat Ha’mazon overrides the Rabbinic edict forbidding reading by candlelight on Shabbat, and thus he may read the text in order to fulfill the Misva. Hacham Ovadia Yosef rules that a person in this case may read the entirety of Birkat Ha’mazon by candlelight, including the fourth Beracha, which is required only by force of Rabbinic enactment.
Another exception, which is mentioned by the Shulhan Aruch, is that one may pray by candlelight on the night of Yom Kippur, despite the fact that all Shabbat prohibitions apply on Yom Kippur. The Shulhan Aruch explains, "Emat Yom Ha’kippurim Alav" – a person senses the special awe and reverence of Yom Kippur, and will not mistakenly tilt the lamp. Therefore, it is permissible to pray by candlelight on Yom Kippur. However, the Halachic authorities limit this rule to Yom Kippur. If Rosh Hashanah falls on Shabbat, one may not pray by candlelight.
Given that the Shabbat candles have a special status of sanctity, it is forbidden to walk in their presence when one is not properly dressed, or to perform disrespectful activities by their light, such as changing an infant’s diaper. The Poskim warn that treating the Shabbat candles disrespectfully can pose the risk of danger, and thus one must ensure to avoid performing unbecoming activities in the presence of the Shabbat candles.
Summary: Although it is generally forbidden to read by candlelight on Shabbat, it is permissible to read a text which one more or less knows by heart. It is also permissible to recite Birkat Ha’mazon by candlelight on Shabbat. Praying by candlelight is permissible on Yom Kippur, but not on other holidays when they fall on Shabbat. One may not walk around undressed or perform disrespectful activities – such as changing a baby’s diaper – by the light of the Shabbat candles.