How can I effectively save tax with a salary of over 15 LPA in India?

While the only option to get that figure down to zero is to leave your
job, other viable options which will give you maximum benefit includes:

(I have included a personalised sketch of deductions crafted according to the facts of your case at the end.)

1. NPS
New Pension Scheme is a relatively new introduction to income tax deductions which can be exploited to our maximum advantage.

• Eligibility- Both employed as well as non-employed individuals.

• Contribution- Minimum limit is Rs. 600 per annum. In case of employed individuals, employer will also contribute same amount.

• Fund- Your fund will be managed by one of the six fund managers--LIC, UTI, Reliance mutual fund being three of them--appointed by the Government. You can also prescribe the proportion in which you want your funds to be invested in equity, govt. securities, fixed income securities respectively.

Now coming to point of our interest, deduction. Deduction is available under two sections.

1. 80CCD(1)
Deduction is availabe upto
10% of ( Basic + DA )
In your case it comes to Rs. 150000 (assuming your 15L salary is exclusive of any perq.)

But, there is a complication, another obscure section of income tax act places a limit:

[ 80C + 80CCD(1) ] <= 150000

From financial year 2015-16 (this year) additional Rs. 50000 is available over and above the above 150000 limit.

2. 80CCD(2)
Now, the beauty of this section, you can also get deduction of your Employer's contribution.
Deduction is least of the following two:

• 10% of ( Basic + DA ) i.e Rs. 150000.
• Rs. 1 Lakh.

1 lakh is the limit. So you will be able to avail 1 lakh deduction in addition to the above mentioned deductions. 

2. Medical insurance premium u/s 80D.
Under this you can get deduction upto Rs. 20000 on your mediclaim premium.

You have already taken Life Insurance policy and opened a PPF account. You might have to change amounts you invest in PPF and choose a different policy to streamline with NPS and claim max benefit.

You can also look into other options which involve investment in bonds and donations to some specified institutions. 

Optimal deduction structure:

80C
Lic premium/ PPF - Rs. 50000
(This will require some change in your original holding pattern.)

80CCD(1)
Rs. 150000 + extra Rs. 50000
Total Rs. 200000

80CCD(2)
Rs. 100000

80D
Rs. 20000

80TTA
Rs. 10000

Other odd 20000/30000

Total Rs. 4 Lakhs.

Tax will come around Rs. 13500 p.m. , Rs. 1500 less than your current tax expense.

Total savings of Rs. 18000 p.a.