My theory is...

a) That Japanese and Tamil are sister languages, even though they do not share close vocabulary correspondences in their current forms [I].

b) That there was once a Proto-Japonic-Dravidian language (PJD), that later diverged into Proto-Japonic (or Proto-Japonic-Ryukyuan), and Proto-Dravidian. Proto-Dravidian later diverged into the languages of southern India (including Tamil, Malayalam, Telugu and Kannada). Proto-Japonic (or Japonic-Ryukyuan) diverged into the modern day Japanese (and perhaps other Ryukyuan languages) [II].

c) That over the centuries, Japanese was strongly influenced in turn by other language groups: Old Korean, Austronesian, Altaic and Chinese and ended up absorbing a lot of their vocabulary. However, it largely retained its native grammar, which it inherited from Proto-JD [III].

d) That a nearly similar fate befell Tamil. With the growing influence of Sanskrit in northern India, and the increasing patronage of Vedic Brahmins by southern Indian rulers [IV], Tamil began to absorb a huge corpus of Sanskrit words, while retaining its underlying grammar (which remains distinctly different from Sanskrit even today).

e) That, if one should look for vocabulary correspondences, one should look not in old/classical texts (either of Japanese or Tamil) but in the regional dialects in the heartlands. In an attempt to standardize the Japanese language (and weed out native dialect differences) during the time classical texts were being written, it is possible that Japanese rulers imported several Chinese words into Japanese [V]. So vocabulary correspondences between Japanese and Dravidian, if preserved at all, would be preserved in local dialects, and not the standardizes (textual) form of the language.

Clearly, a lot of this is speculation, but in the next sections I will present evidence to lend weight to Japanese-Dravidian hypothesis.

Footnotes:

[I]: Although they do share very similar (agglutinative) grammars, with very strong word-order similarities, as noted before.

[II]: There exists the possibility of a Proto-Japonic-Altaic-Dravidian language, but as I have no knowledge of any Altaic language, I cannot bring any new ideas to that hypothesis.

[III]: It is well-established that Chinese and Japanese have completely different grammars, and are not part of the same language family [1]. For further evidence, see next section (Japanese and Chinese)

[IV]: The Pallava kings (c. 600CE, date uncertain) wooed Brahmins from northern India with land grants and settled them into the Kaveri basin in southern India [12].

[V]: China has been the economic and intellectual powerhouse of East Asia over a lot of history. So it is plausible that the Japanese rulers used Chinese words (instead of a native dialect) to standardize the language for texts (after all, the writing system in these texts is the Chinese system!).

This situation is not without parallel in history. The Mughals who ruled India (15th-18th centurey CE) were essentially of Turko-Mongol origin (the word Mughal is, in fact, cognate to the word Mongol). Now, both Turkic and Mongolic belong to the Altaic family of languages. However, the language of the Mughal court and a lot of beatiful literature composed at this time was neither the language of the rulers (Turkic/Mongolic) nor the language of the natives (Sanskrit/Prakrit). Instead the Mughals adopted Persian, an Indo-Iranian language, and (ironically) a sister language of Sanskrit, which was the lingua franca in adjoining Persia, the political and intellectual powerhouse of the near East under the Safavid dynasty in the 15th century [14,15]. 

Japanese & Chinese: Not all that looks like Chinese is Chinese

Although Japanese writing shares a lot a common with Chinese, Japanese and Chinese are, in fact, completely unrelated languages! Japanese has two (in fact three) scripts: (a) the Chinese-borrowed script (Kanji), with related sounds, to represent vocabulary/ideas (such as verb roots), and (b) (Kanji-derived) Kana to represent the grammatical conjugations specific to Japanese. Each Japanese word is, thus, a composite of a Chinese kanji (to express a core idea) with Japanese-specific grammar endings (written with Kana) tacked on.

For example: 食べられなかった (taberarenakatta) means "Could not eat". However the root of the idea, eat, is the Chinese-borrowed Kanji 食 (ta). The rest of the characters are Kana suffixes indicating the mood or tense of the word. In this example, られ (rare) indicates potential/ability (can), な (na) indicates negative (not), かった (katta) is a past tense marker. There is no equivalent grammatical structure in the Chinese language.

But why did Japanese assimilate only Chinese vocabulary and not the entire grammatical structure of the language? Perhaps because grammar is more resilient to change than vocabulary (any linguists?). To take a simple example, a child learning a foreign language often tries to add the vocabulary of the new language (as superstratum) over the basic structure of his/her native language grammar. For instance, native Tamil speakers not proficient in English can be heard to say "why you are leaving?" (instead of "why are you..."). This is probably because in Tamil the word that naturally follows the question word ("why") is the subject of the sentence ("you") [The situation is actually, a little more complicated, because modern Tamil has lost the copula ("are"). However, even if it were to appear, the copula would appear at the end of the sentence exactly as in Japanese, and as in modern Malayalam (which has retained the copula)].


Next section: Linguistic parallels