The gemara (Shabbos 23) presents a case of someone who lit a very long-burning candle on Erev Shabbos e.g. a yahrzeit type candle, that was still burning Motzei Shabbos. Since hadlakah oseh mitzvah, to fulfill the mitzvah of ner chanukah the candle must be extinguished and relit. Rashi and Tosfos have slightly different readings of the case, but those slight differences may have major significance. Rashi explains the case being addressed is a long-lasting Chanukah candle lit on Friday afternoon – if that same Chanukah candle is still burning Motzei Shabbos, it must be extinguished and relit. Tosfos explains the case being addressed is long-lasting Shabbos candles lit on Friday – if the Shabbos candles are still burning on Motzei Shabbos, they must be extinguished and relit for the sake of Chanukah.
The Rogatchover infers from Tosfos that the need to extinguish and relight the candles is only where the candle was originally not lit for Chanukah, e.g. Shabbos candles. But if one lit a Chanukah candle and it remained burning through the next night, it would not have to be relit, contrary to Rashi's view.
The Rogatchover poses the following chakira: is Chanukah 8 one-day holidays, or 1 holiday lasting eight days? If it is 8 individual holidays, then each night needs its own lighting, as Rashi holds. However, if Chanukah is a single 8 day unit, then theoretically a candle lit for Chanukah which burned for all 8 days would suffice, as Tosfos holds.
This machlokes between Rashi and Tos. May have other ramifications. If a child becomes bar mitzvah or a ger converts in the middle of Chanukah, according to Tosfos there may be no chiyuv to light menorah – if one is patur at the start of the eight day holiday, then one is patur for its duration. Only if one considers each night a separate Yom Tov, like Rashi, would a new chiuv take effect.