It shouldn't. Honestly, you probably don't even need the ram heatsinks any more. They are just a holdover from DDR2 when the Ram did need the extra heat dissipation. DDR3 runs cooler, but since people are used to seeing the ram sinks, they don't feel comfortable buying ram without it.
However, you'll be getting air flow over the ram with the fans from your cpu cooler, which will keep it cooler and there is no reason to think that it would get warm enough to melt the cable.