I recently had to renew my passport. The expired one has an Indian visa that will be valid until 2016. Is it necessary for me to get the visa transferred to the renewed passport, or can I get through customs in India (likely Mumbai) if I have both the expired and the renewed passports with me?
Yes, you will simply need to present both passports and you will be fine, as long as a few simple conditions are met.
This is the specific information from TIMATIC (the visa service used by most airlines/etc) :
Presuming none of the items above are issues, just present both passports and explain the situation and you'll have no problems.