You might be embarrassed to discuss this with your father, but unfortunately, you're going to HAVE to if you're going to have a chance to do anything about the problem. Don't worry so much about it, though. He's your father and he loves you, and if you tell him sincerely that this problem is embarrassing you and is negatively affecting your life, he will be able to relate and he will want to do what he can to help you. Ask him to imagine going through his high school years with such a problem.....never even feeling comfortable to remove his shirt in public. Any dad will be able to sympathize.
Yes, you are too heavy and need to lose some weight. But it is not likely that the weight loss alone will do much to help your puffy nipples.
I suggest telling your dad and maybe you can even have him read some things on this site so he will get a better understanding of the problem. Then, ask if he can look into getting surgery for the problem, which will require you to find a board certified plastic surgeon with a lot of experience at correcting gyne.
This is something you need to have corrected as soon as possible, in my opinion. Usually, summer is the best time for guys who are in school. This should be one of the most fun periods of your life. You need to be able to live normally and feel good about yourself. So make it happen.