It seems like India with all the different beliefs would by chance maybe have something? Would Muslims and Hindu be open to such an idea? A Mosque combined with a Temple?
Could be pretty popular?
Hindu/Muslim
Muslims hate the following things which are commonplace in Hindu temples: statues, singing, women and men mixing. Probably other things.
This could be a problem.
Hindu/Sikh,Buddhist, et cetera
I think the objections here would come from the minority Indian faiths who would be scared of the erosion of their claim to have traditions separate from the large multiplicity covered by the term "Hindu". somewhat of an Identity issue.
Hindu/Christian
In fact some missionaries have tried to blend these elements as a way of assisting the conversion of Hindus to Christianity. This is obviously the opposite of what you were suggesting and natural suspicion of Christian motives would I think stop anything approaching what you suggest from developing.
Also there would be caste issues, but I mention them last because I don't think they would have as much impact as the other issues.