The basic elliptical trajectory wouldn't be affected at all by the sun's gravity, that's what it is all about. The atmospheric drag and solar wind should cause a noticable effect. Comet Lovejoy is a member of a class of comets called Kreutz sungrazers. They are the remants of an 11th Century comet that was broken up by tidal forces when it passed too close to the sun. The solar corona is about a trillion trillion times less dense than the earth's atmosphere at sea level, btw.
It didn't survive because the sun is cooler than they thought but because Lovejoy is bigger than expected. It will not survive many such approaches.