It looks like this version of Unicode was released in March 2005. This means that Tamil fonts created before, such as the default one in Windows XP, will not have this character. If you are interested, you can download the "Lohit" tamil font, which supports the new code-point. It should work on Windows and Linux. (Not sure about Mac.)
As I feared, there were some anti-Sanskrit folks who fought against this character. From a Unicode mailing-list thread on this subject:
Sanskrit is always seen a wanton intrusion [sic] to destroy all Indic languages and cause confusion. Tamil has been defending itself for hundreds of years.... Unicode is not the entity that should decide the demise of the ancient and sophisticated Tamil, like the demise of all other Indic languages.... 0BB6 must be deprecated. 0BB6 was encoded illegally by Unicode.
I am pleasantly surprised that, inspite of this, 0BB6 made it through.
Right now in translipi, I use ச to transliterate 'sha' into Tamil. This is fine for the cases where it is accompanied by a vowel. However, when it appears as a pure consonant or together with other consonants, ச is conventionally pronounced 'cha' in Tamil. This is not very satisfactory; the grantha 'sha' letter fits the bill perfectly here. However, since not many fonts support it yet, translipi will go along with ச for 'sha' for some more time.