According to the folks at On Rope 1, "Not unless you are wearing ice crampons or razor blades.  There is even recent discussion that claims crampons in snow just pass through the rope fibers and cause no harm to the rope."

They go on to report:

"Jim Kovach, in an exhaustive 8 month study in Ohio, and reported to/by ITRS (International Technical Rescue Symposium) including using rope as a doormat for months, and then driving over the same rope while it lay on beds of broken chards of bricks and a glazier's pile of broken glass. Then the rope was pull tested! Although the kernmantle showed minor signs of wear, in testing no loss of strength was seen. There is no evidence that stepping on a rope will grind dirt past the kernmantle into the core."

Perhaps equally interesting: 

"In other testing done by CMC out of California, rope were immersed in various substances for long periods of time and then tested. Some of these included paint, tar, sap, gasoline, oil and vomit. Most of the test items, including gasoline and oil, did not cause strength problems (although an oily rope is tough to use, it did not weaken the rope). Acidic materials, like vomit and animal urine, can cause a 30 percent or more strength reduction."

So the bottom line:  It's more dangerous to have your packaged patient puke on your rope, or for a SAR Dog to lift a leg on your rope than for you to walk on it.