I am looking for sources that discuss the idea of paying tuition money ("schar limud") towards obtaining a Torah education. Topics of interest include: The importance of paying tuition for Torah; importance of timely paying; assurances/promises that paying such money will not be harmful financially etc. Sources from any area in Torah (Chazal, Rishonim, Achronim, Mussar, Chassidus) are welcome.
The last question first, the Gemara in Beitza 16a says that the more one spends for Shabbos & Yom Tov and for expenses of Talmud Torah for his children the more he receives [from Heaven].
I would suggest to learn the first chapter of Hilchos Talmud Torah in Shulchan Aruch Horav, from Halacha 2-9 he elaborates on the obligation of the father, grandfather, and Bais Din to pay for learning Torah and when we force the father to pay etc., with excellent Mar'ei Mekomos.
He doesn't talk about timely payments, but it would be the same as any hired worker which has the positive commandment "Pay him his payment in his/its day" and the negative commandment of "Don't delay the wages of the worker".
Technically, one cannot receive money for teaching Torah:
"ראה למדתי אתכם חוקים ומשפטים כאשר צוני ה' אלקי" – מה אני לימדתי אתכם בחינם, אף אתם כשתלמדו זה לזה – בחינם.
A teacher was able to take money for one of two reasons - as payment for watching kids, or as "sechar batala" for not doing something else during that time. Once there's a heter to take money for teaching torah, one can discuss the chiyuv to pay for it.
Originally, a father would fulfill his obligation of "ולמדתם אותם את בניכם" by teaching his son Torah (i.e. Mikra), so there was no financial obligation. R. Y. b. Gamla instituted that there should be teachers everywhere for children to learn Torah. (Baba Basra 21a)
דאמר רב יהודה אמר רב ברם זכור אותו האיש לטוב ויהושע בן גמלא שמו שאלמלא הוא נשתכח תורה מישראל שבתחלה מי שיש לו אב מלמדו תורה מי שאין לו אב לא היה למד תורה מאי דרוש (דברים יא, יט) ולמדתם אותם ולמדתם אתם התקינו שיהו מושיבין מלמדי תינוקות בירושלים מאי דרוש (ישעיהו ב, ג) כי מציון תצא תורה ועדיין מי שיש לו אב היה מעלו ומלמדו מי שאין לו אב לא היה עולה ולמד התקינו שיהו מושיבין בכל פלך ופלך ומכניסין אותן כבן ט"ז כבן י"ז ומי שהיה רבו כועס עליו מבעיט בו ויצא עד שבא יהושע בן גמלא ותיקן שיהו מושיבין מלמדי תינוקות בכל מדינה ומדינה ובכל עיר ועיר ומכניסין אותן כבן שש כבן שבע
The Aruch HaShulchan (יורה דעה · סימן רמה) discusses further details. People who could afford it would hire their own tutors for their sons. Everyone had to pay some kind of tax to support "Talmud Torahs" where poor kids and orphans would learn. If someone wealthier wanted to send his son to the Talmud Torah, he would have to pay extra tuition on top of the general tax he already payed.
We can force people to pay for their son's education and for the communal education. If a father cannot pay, we can even force the grandfather to pay for the education of his grandson.
Although the main chiyuv is to teach his son mikra, the Tur and Shulchan Aruch (245:6) write that someone who can afford it should pay to teach his son Mishna and Gemara also:
היה מנהג בעיר שלוקח מלמד תינוקות שכר – חייב ללמדו בשכר עד שיקרא תורה שבכתב כולה. ואינו חייב ללמדו בשכר משנה וגמרא. והני מילי דלא אפשר, דדחיקא ליה שעתיה. אבל אם אפשר לו – מצוה לאגמוריה משנה וגמרא, הלכות ואגדות.
(It doesn't seem like everyone continued learning in school when they were teenagers.)
In summary, there is a chiyuv on the father to pay to teach his son torah and to support the education of the poor also.
As mentioned by @Meir, the obligation to pay on time should be similar to the obligation to pay any worker. One cannot delay payments to an institution either.
Also mentioned, רב תחליפא says (Beitza 16a):
כל מזונותיו של אדם קצובים לו מראש השנה ועד יום הכפורים חוץ מהוצאת שבתות והוצאת י"ט והוצאת בניו לתלמוד תורה שאם פחת פוחתין לו ואם הוסיף מוסיפין לו
(Though with today's tuition costs, one wonders if רב תחליפא would say the same thing today!)
According to the sources I've seen, a person who finances the study of Torah by others certainly is rewarded in the world to come with a share of the scholar's merit. In essence, this is a contractual relationship. Although a student cannot sell his entire reward for learning in exchange for financial support, mostly because it would be unconscionable to do so, Rabbi Yerucham, and others, say that a Torah scholar can give a share of his reward to his sponsor. Rabbeinu Yerucham Ne/hil' 2 to end. But if he does so, what he gives away is subtracted from what he earned, says Rabbi Joseph Karo, author of the Shulchan Aruch. Responsa Avkath Rokhel 2. ("[The one who studies all day] may give half his reward for Torah study to another... and it is then as if he studied Torah for half a day"); accord ShaKb, SA Yoreh De'ah 246, note 2.
According to various midrashim the tribes of Zebulun and Issachar had this sort of contract -- Zebulun worked in business and supported the tribe of Issachar who were purely devoted to learning. E.g. Midrash Rabba Genesis 99:9. The midrashim cite this contactual relationship as an explanation as to why the Torah lists Zebulun ahead of Issachar despite the fact that Issachar is older -- Zebulun advances based upon his support of Torah study.
Nowhere in the Talmud is the Zebulun-Issachar contract recognized one way or another as a good or bad idea. If anything, the Tannaim would appear to have been against the concept of full-time, subsidized scholarships for Torah students. Mishna Avos 2:2 says: "Any Torah not associated with labor will end in failure and cause sin."
The sages clear preference is that all Jews should study Torah and not trying to get credit for other people's learning by simply giving them money to learn. But, the rabbis also understood that for some people, sitting down and learning every day is not an option. We see this in a discussion by the rabbis of the last curse of the Tochecha -- the list of curses given to the Jewish people as a warning should they forget to observe the Torah -- which says: "Cursed be he who will not uphold the words of this Torah in order to fulfill them." Deuteronomy 27:26. What did the Torah mean by "uphold the words of this Torah"? The Yerushalmi Talmud concludes that: "[Even] if one learned, taught, observed and fulfilled, but had the opportunity to strengthen [the observance of others] and did not strengthen it, he is included in the curse." JT Sotah 7:4. Accordingly, in their view, it was not enough to study Torah, you had to help others to do so as well. Another view notes that there is no explicit order in the verse to imply that "upholding Torah" means that one MUST study Torah, Rabbenu Yonah says that means "[o]ne should observe the actions of his fellow workers [in the service of God]" and encourage them by giving financial support. Sha'arey Teshuvah 3: 19.
I think one of the most immediate rewards is obtaining nachas from the learning of someone you supported. On this topic, one of the most meaningful stories is the story of Rabbi Akiva and his wife Rachel. Rachel gave up her relationship with her father (and possibly an inheritance -- that is less clear) and worked for 24 years while her husband lived away from home and studied Torah. The Gemara at Nedarim 50a tells the story of Akiva's triumphal return home with his 24,000 students. Rachel tries to get up close to welcome her husband home. But because she is wearing rags and has nothing better, the students ridicule her, asking how she could come dressed like that to see a great sage. Akiva tells his students, "all that I have [i.e. his Torah learning] and all that you have is from her." Whether or not any of his learning merited her a place in the world to come, the measure of nachas she must have felt, I think, made her sacrifices worthwhile to her.