Gemara

(Mishnah): If a camel laden with flax was walking in a Reshus ha'Rabim, and the flax entered a store, caught fire from the storeowner's lamp and burned the building, the camel's owner is liable.

If the storeowner left his lamp outside, he is liable.

R. Yehudah says, if it was lit for the Mitzvah of Ner Chanukah, he is exempt.

Shabbos 21b (Ravina): From R. Yehudah, we learn that Ner Chanukah must be placed within 10 Tefachim of the ground.

If it was permitted to place it higher, the storeowner should be liable. He should have put it higher than a camel and its rider, to avoid damage!

Rejection: (Really, one may put it above 10.) Chachamim did not obligate one to place it so high, lest he refrain from the Mitzvah.

(Rav Kahana): If Ner Chanukah was placed above 20 Amos, this does not fulfill the Mitzvah, just like a Sukah or Korah (a beam to permit carrying in an alleyway) more than 20 Amos high is invalid.

Rishonim

Rif and Rosh (Shabbos 9b and 2:5): If Ner Chanukah was placed above 20 Amos, this does not fulfill the Mitzvah, just like a Sukah or Korah.

Ran (DH Omar Rav): Above 20 Amos, people cannot see it, so it does not publicize the miracle.

Rambam (Hilchos Chanukah 4:7): If Ner Chanukah was placed above 20 Amos, this does not help at all, for it is not noticeable.

Magid Mishneh: The Gemara disqualifies above 20 Amos. This shows that anything below, even above 10 Tefachim, is Kosher.

Rosh (ibid.): Even though Ravina's proof was rejected, the Halachah follows him. It publicizes the miracle more when it is lower, for lamps for illumination are not placed so low. Also R. Chananel said so.

Ran (DH Garsinan): R. Chananel says that we do not abandon what Ravina was sure about due to the Gemara's possible rejection. Also R. Yonah and the Rashba said so.

Mordechai (Shabbos 266): Nowadays that we light in the house (lest Nochrim cause problems), we are not particular if it is above 10 for there is (no - Rema deletes this) publicity.

Poskim

Shulchan Aruch (OC 671:6): One leaves Ner Chanukah above three Tefachim.

Beis Yosef (DH veha'Ram): Maharam me'Rotenburg used to leave it between three and 10 Tefachim, for within three of the ground is like resting on the ground.

Mishnah Berurah (26): Within three, it is not clear that the owner put it there. In any case, b'Di'eved he was Yotzei.

Kaf ha'Chayim (45,46): Below three, people do not notice it. According to Kabalah, it should be between three and 10.

Shulchan Aruch (ibid.): The Mitzvah is to leave it below 10 Tefachim.

Beis Yosef (DH v'Chosav ha'Ran): Also Semag and Semak say that one should leave it below 10. The Rif and Rambam omitted this. They hold that since Ravina was rejected, the Halachah does not follow him. Practically, one must leave it below 10, for many opinions require this, and the Rif and Rambam hold that the Mitzvah is below or above 10. Even though the Mordechai says that since we light inside, it makes no difference to us, the Tur did not distinguish. Meticulous people put it below 10.

Mishnah Berurah (27): Eliyahu Rabah says that people are not careful about this because they rely on the Mordechai. Meticulous people are stringent, like the Beis Yosef. One who lives in an Aliyah (upper story) leaves it in his window, even if the window is above 10 from the floor of the Aliyah. If one has two windows and one is within 10 of the floor, surely he leaves in that one. The custom is to leave the Chanukiyah in the Beis ha'Keneses in a high place.

Shulchan Aruch (ibid.): If he left it above 10 Tefachim he was Yotzei.

Gra (13): The Gemara disqualifies above 20 Amos. This shows that above 10 Tefachim is Kosher.

Shulchan Aruch (ibid.): If he left it above 20 Amos he was not Yotzei.

Question (Tur): R. Yo'el says that above 20 Amos is Pasul outside, but in the house it is Kosher, just like a Sukah above 20 is Kosher if the walls reach the Sechach, for then people see it. I disagree. The walls enable one to see the roof. Here, one must be able to see the Neros. Seeing the roof does not help!

Answer (Beis Yosef DH v'Chosav ha'Rav): Mahari Avuhav says that R. Yo'el holds that (open) air impairs vision. If so, in a house also a Ner above 20 is visible.

Rebuttal (Taz 5): If so, why did Rava need to says that a Sukah above 20 is Kosher when the walls reach the Sechach? It should suffice if there are (even small walls below and) walls from 20 Amos until the Sechach! If air below 20 Amos were a problem, we should disqualify a Sukah even less than 20 if the walls do not reach the Sechach! Rather, the walls help to see the roof only if there is no gap. Also, the Mitzvah of Ner Chanukah is after Tzeis ha'Kochavim, and then walls do not help to see the Ner, for also the walls cannot be seen!

Kaf ha'Chayim (48): If the window is more than 20 Amos above Reshus ha'Rabim, others cannot see it, so he leaves the Chanukiyah at the door to the house. We are primarily concerned that people in the house see it, so it suffices if it is within 20 of the floor of the house. Perhaps even so, it is better at the door.

Kaf ha'Chayim (52): We are concerned that the flame be within 20 Amos, and (l'Chatchilah) 10 Tefachim.

Rema: If it was lit above 20 Amos and one lowered it, he was not Yotzei, for the Mitzvah is lighting it.

Source (Tosfos 22a DH Ner, cited in Beis Yosef DH u'Mah she'Chosav Aval): Rather, he must extinguish it, lower it and light it again.

Mishnah Berurah (29): He must bless again.

Kaf ha'Chayim (53): It seems that if he realized before diverting his mind from the Mitzvah, he need not bless again.

Gra (14): The Gemara did not say that if it is above 20 one should lower it, for this would imply that one may do so while it is lit.