It takes longer than a few days or a week for heel drops (also called "heel dips") to have an effect on your Achilles tendon. The Achilles is notoriously slow in adapting to training--the tissue simply doesn't repair at the same rate as muscle (one 2013 study, looking at the levels of carbon-14 in the muscles and Achilles tendons of subjects who'd lived through nuclear bomb testing from 1955 to 1963 found that while muscle had cleared all the carbon-14, the levels in the tendon hadn't changed, giving you an idea of how quickly Achilles tendon tissue regenerates itself when left to its own devices).
I do heel drops a couple times a week (3 sets of 20 reps, holding a 20 lb dumbbell in the same-side hand--although a weight vest would work better). I used them to recover from an Achilles insertion tear that had lingered for 6 years and led to a point where I couldn't run more than a mile or two without stopping (I'd walked with a severe limp for all 6 years). After 7 months of drops, I was able to run, sprint, hill-sprint, race, do drills that including bounding, hopping, etc. No pain. No limp. And back up to 85 miles/week. Of course, I did a few other exercises, too, to improve range of motion, stability, and strength throughout my lower body, but it's the dips that provided the magic.
Those on this thread who are experiencing pain or negative consequences from heel drops might be doing them incorrectly. They shouldn't be impacting your calves almost at all. If they are, you might be doing them too slowly. Or maybe you're treating them like calf raises, too? In any case, all you need to do is control the lowering of the working heel--not slow it to the point that you're working your calf--and then use both feet to quickly rise back up on your toes. Of course, the exercise won't work for everyone, but it's the best remedy out there (so far).
To the poster who wanted links, here's a good one, with plenty of other links available at the end of the study:
http://bjsm.bmj.com/content/46/3/214