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A brief essay showing that the earnings one makes from teaching the Qur‘aan, is from the best forms of attaining sustenance.
My people know that my profession was not capable of providing substance to my family. And as I will be busy serving the Muslim nation, my family will eat from the National Treasury of Muslims, and I will practice the profession of serving the Muslims.
This topic was also discussed in the third book along with its proofs. It is appropriate for him to teach without taking wages if he is self-sufficient. If he is in need, then his salary is from the Baytul Maal because teaching is part of the general benefit to the Muslim public. If it was difficult to get money from the Baytul Maal, and the people in his district or village agree to give him a salary , then that is permissible. It is also permissible to receive money for teaching someone.
The proofs for all these are:
1) It is more appropriate to teach voluntarily, without a salary.
a) Conveyance of knowledge is an Islaamic duty like Salah and fasting. Allaah said,
"(And remember) when Allaah took a covenant from those who were given the Scripture (Jews and Christians) to make it ( the news of the coming of Prophet Muhammad and the religious knowledge) known and clear to mankind, and not o hide it, but they threw it away behind their backs, and purchased with it some miserable gain! And indeed worst is that which they bought." (Soorah Aali-'Imraan 3:187)
Therefore, he does not deserve a salary because it is obligatory on him.
This tafseer is in Qurtubee (1/185).
b) Taking the example of the Prophet (sallallaahu 'alayhi wa sallam).
"He should follow the example of the Prophet (sallallaahu 'alayhi wa sallam) by not taking a reward for anything he teaches the people. He should not expect compensation or gratitude, but he should teach for the sake of Allaah and to draw nearer to Him. He should not perceive that he is doing a favour to them, even though gratitude on their part is a must. On the contrary, he should notice their virtue of purifying their hearts to draw nearer to Allaah and allowing him to plant his knowledge there. Just like the one who loans you his land to cultivate. The benefit you receive from his land is more than his. So how do you burden him for a favour when your reward with Allaah is more than the students' reward? If it was not for the students' learning, he would not receive this reward. So do not ask for a reward except from Allaah.
"And O my people! I ask of you no wealth for it, my reward is from none but Allaah." (Sooratul Hood 11:29)" 
2) He should take a grant from the Baytul Maal if needed. Because he is doing a duty to the Muslims which is fardh kifaayah (obligatory upon the Muslims as a whole), his sustenance comes from the Baytul Maal.
Al-Baghdaadee said, "It's the Imaam's duty to allot certain wages from the Baytul Maal for the ones who teach and gives fatwaa, so they can focus their time on teaching, instead of earning a living." 
The proof for that is that the Companions (radiyallaahu 'anhum) did it with Abu Bakr when he was Caliph.
Narrated 'Aa`ishah, When Abu Bakr was chosen Caliph, he said,
"My people know that my profession was not capable of providing substance to my family. And as I will be busy serving the Muslim nation, my family will eat from the National Treasury of Muslims, and I will practice the profession of serving the Muslims."
Ibn Hajar said,
"The designated amount was agreed upon by the Companions."
Ibn Sa'ad narrated with a mursal isnaad, whose men were of reliable character:
"When Abu Bakr was made Caliph he went to the market with some thawbs to make business. 'Umar and Abu 'Ubaydah saw him with the goods. They said: 'You're doing that and you're the Leader of the Muslims?' He said; ' How can I feed my family?' They said; 'We will assign to you half a sheep for each day."
3) If there is no Baytul Maal, and he is in need, he should take a fee from the village-people or from his student.
Narrated Ibn 'Abbaas from the Prophet (sallallaahu 'alayhi wa sallam),
"It is perfectly legal to charge for Allaah's Book." (Qur`aan).
Ibn Hajar said,
"The majority of the scholars use this Hadeeth to support taking a salary for teaching The Qur`aan. The Hanafees contest that. They say it is not allowed to charge for teaching but it is allowed for healing by Ruqyah. They also say teaching the Qur`aan is an act of worship, and the reward for worship is from Allaah."
Hanafis also have more proofs for their ruling. Of it, is this statement and all others with the same meaning:
"And believe in what I have sent down (this Qur`aan), confirming that which is with you, (the Torah and the Gospel), and be not the first to disbelieve therein, and buy not with My Verses (the Torah and the Gospel) a small price, and fear Me and Me Alone." (Sooratul-Baqarah 2:41)
They also used a lot of weak ahaadeeth to support their views. Qurtubee mentioned the weak ahaadeeth that they use in his tafseer. Refer to it if you wish.
Ibn Taymiyyah said,
"The scholars of Islaam have a disagreement on the issue of taking recompense for the Qur`aan. There are three different opinions…the most correct one is that itis permissible for the one in need." 
If this ruling, charging for teaching the Qur`aan, is established, then it is also permissible to receive wages for teaching other Islaamic studies.
 Tafseerul Qurtubee (1/185)
 Al-Ghazzaalee, Ihyaa 'Uloomud-Deen (1/70)
 Al-Khateeb Al-Baghdaadee, al-Faqeeh wal-Mutafaqqih (2/164)
 Related by Bukhaaree (3/284)
 Ibn Hajar, Fathul - Baaree (4/305)
 Related by Bukhaaree (3/263)
 Ibn Hajar, Fathul-Baaree (4/453)
 Tafseerul Qurtubee (1/355)
 Ibn Taymiyyah, Majmoo'ul-Fataawaa ( 30/192)