mezuzah - Source that every room in the house needs a mezuza

What are the source(s) that every room needs a mezuzah and not just the front door?

In the Torah and Talmud.

Answer

There does not appear to be a Talmudic source that explicitly states that every room in a house requires it's own mezuzah. Rambam, however, does explicitly state this:

Hilchot Mezuzah 6:10

חדר שבבית אפילו חדר בחדר חייב לעשות מזוזה על שער החדר הפנימי ועל שער החדר החיצון ועל שער הבית שכולן עשויין לדירה וקבועין

When there is a separate room in a house, or even one room which leads to another room, it is necessary to affix a mezuzah on the doorway to the innermost room, the doorway to the outer room, and the doorway to the house, since all of them are used for the purpose of dwelling and are permanent structures. (Touger translation)

The subsequent major codes all adopt Rambam's ruling (Tur Y.D. 286, Shulchan Aruch Y.D. 286:16, Levush Y.D. 286:16, Aruch Hashulchan Y.D. 286:37. The respective commentaries that provide the sources for those codes send the reader right back to the statement of Rambam quoted above.

The question then is, what was Rambam's source?

R. Yosef Karo in his commentary thereto wonders why Rambam had to include this rule at all, since it is already covered by a previous rule. He does not tell us which previous rule he is referring to, but the other commentaries appear to assume that he is referring to the following:

Hilchot Mezuzah 6:8

לפיכך אחד שערי חצרות ואחד שערי מבואות ואחד שערי מדינות ועיירות הכל חייבים במזוזה שהרי הבתים החייבין במזוזה פתוחין לתוכן אפילו עשרה בתים זה פתוח לזה וזה פתוח לזה הואיל והפנימי חייב במזוזה כולן חייבין ומפני זה אמרו שער הפתוח מן הגינה לחצר חייב במזוזה

Accordingly, gates to courtyards, gates to alleys, and gates to cities and towns, all require a mezuzah, since houses which require a mezuzah open up to them. Even when there are ten structures leading one to each other, should the innermost one require a mezuzah, they all require [mezuzot]. Therefore, [our Sages] stated: A gate which opens up from a garden to a courtyard requires a mezuzah. (Touger translation)

In his commentary to that ruling, R. Karo traces the law back to the following Talmudic statement:

Menachot 33b

אמר רחבה אמר רב יהודה בי הרזיקי חייב בשתי מזוזות מאי בי הרזיקי אמר רב פפא סבא משמיה דרב בית שער הפתוח לחצר ובתים פתוחין לבית שער

Rehabah said in the name of Rab Judah, An entrance-lodge requires two mezuzoth. What is meant by ‘an entrance-lodge’? — R. Papa the Elder said in the name of Rab, It is a lodge, with one door opening on to the courtyard and another leading to the dwelling-houses. (Soncino translation)

Thus according to R. Karo, the source that every room in a house requires a mezuzah is a bit of a leap. We start with a Talmudic statement that when there is an entrance structure between the main structure and the courtyard, the entrance structure requires a mezuzah in addition to the main structure's mezuzah. This is then expanded to include even a chain of ten such structures, which in turn is taken to mean that any room within a structure requires a mezuzah.

That this is a bit of a leap is noted by some of the commentaries. R. Yedidya Shmuel Tarika argues that the cases in 6:8 and 6:10 are not necessarily analogous. In the former, all the outer structures require a mezuzah because they are the entrance structures to an inner structure that requires a mezuzah. In the latter case, though, each room is just another room of the house so it should be covered under the mezuzah by the entrance to the house:

Ben Yedid Hichot Mezuzah 6:10

ואפשר דהו"א דוקא עשרה בתים זה פתוח לזה הואיל והפנימי חייב כולן חייבים אבל בית שיש בתוך הבית עצמו חדר וחדר לפנים ממנו שהכל בית אחד תסגי במזוזה אחת

Somewhat similarly, R. Massoud Chai Rakkah points out that in the former case the outer structures require a mezuzah because they are necessary to get to the inner structure; in the latter case it is the reverse – we would be requiring a mezuzah on an inner room on account of the rest of the house:

Maaseh Rokeach Hilchot Mezuzah 6:10

אפשר לומר דהתם שאני שאף ששאר הבתים החיצונים אין חייבים מצד עצמם מ"מ כיון שהפנימית חייבת והוא צריך ליכנס בה דרך שאר הבתים להכי מחייבו כולם אבל הכא הוי איפכא

Both of them ultimately conclude that the reason why even an inner room would require a mezuzah is simply because it is used for living. Indeed, that is what Rambam himself said at the end of 6:10 – "since all of them are used for the purpose of dwelling and are permanent structures":

Ben Yedid

יהיב טעמא מדעת עצמו שכולן עשויין לדירה וקבועין

Maaseh Rokeach

דכיון שכל החדרים עשוים לדירה אין לחלק בהכי

According to this, then, the rule that every room in a house requires a mezuzah is apparently based on the assumption that the Talmudic ruling which states that the Torah specifically refers to structures that are used for living means that anything used for living, even an individual room that is part of an overall living structure, requires a mezuzah:

Yoma 11b

ת"ל בית מה בית מיוחד לדירה יצאו אלו שאין מיוחדין לדירה

therefore the text reads, ‘house’ — [meaning] just as ‘house’ means a building appointed for a dwelling it thus excludes all other buildings not appointed for a dwelling. (Soncino translation)