Can an American LLC loan money to an individual in India for business
purposes?  What are the legal and tax ramifications?

Hi 

The rules in India about offshore loans  are quite strict. you are
allowed to transfer only upto a max of US$ 250,000 n that too by close
relatives. However, there are some flexiblites as long as they are not
used for some restrictive businesses: Please refer

http://www.indialiaison.com/loans.htm

you can also try sending the moeny as a forigen contributor which is
much easier: Refer
http://www.indialiaison.com/fcra.htm

However, be careful of the dreaded Foreign Exchange Management Act (FEMA)

Which prohibts:
The Reserve Bank can, by regulations, prohibit, restrict or regulate
the following :-

1 )any borrowing or lending in foreign exchange in whatever form or by
whatever name called;

2) any borrowing or tending in rupees in whatever form or by whatever
name called between a person resident in India and a person resident
outside India

Refer :http://www.indiainfoline.com/lega/fema/ch02.html

Hope this helps