I have been on the ATL-BOM route (and the JFK-BOM route before it) several times, and these days Delta usually sticks pretty close to the great circle route. The great circle route indeed crosses the heart of Russia, passes a little north of Iceland, and over the southern part of Greenland. On my last trip home from BOM, we flew over 4 of the six "Stan" family members, including directly overhead Khandahar, Afghanistan. Iran is slightly west of the track, but the actual route varies due to wind and weather, so sometimes the flights do go over Iran. I think overflight of Afghanistan only started within the last year or so, based only on my experience; maybe someone else can correct me if mistaken. Overflight of Iran has been going on for a few years now.

The first few times I went to India, all connections were in Europe, and the flight path crossed over Egypt, Jordan and Saudi Arabia, in a rather tortuous path which seemed designed to avoid Syria, Iraq and Iran. But I haven't been by this route in quite a while; overflight of Iran is much shorter.