How can I hang a heavy frame without making holes in the walls?
This may be thinking outside the box, but there are other ways to display a picture rather than "hanging" them. Hopefully these inspire some solutions...

  

Have you considered using some sort of display easel like the one above instead? 

Google Search: Photo Display Easel

Compression poles are also an interesting way to hang things without causing damage:



Or if you are more industrious, you can construct some sort of free standing (or leaning) panel to hang the picture from. A nicely framed section of painted drywall or other type of surface can make a nice accessory for the room. It can be made free standing or designed to lean up against the wall without causing any permanent damage.