In Hilchos Channukah (OC Siman 675 Sif 1 -- see also there Mishneh Brurah Sif-Katan 3) we learn that if one lit Channukah candle(s) (for the mitzvah) for one day and they stayed lit until the following day, he must nonetheless extinguish the candles and re-light them now for the sake of the mitzvah (of ner channukah) since each day is a mitzvah in and of itself.
What about in regards to Shabbos? That is, if someone lit Shabbos candles Erev Shabbos Parshas Chayeh Sarah (for example) and they stayed lit until Erev Shabbos Parshas Toldos and would easily burn well into the night, is this enough? Or must new candles be lit in honor of this Shabbos? The main reasons for Shabbos candles is for Shalom Bayis — in order that one may see around the house — as well the enjoyment one has by eating by the candles. Seemingly these two things are fulfilled with the candles that remained lit from the previous Shabbos.