There are 80 some species of yarrow. Common yarrow and fern leaf yarrow are the two most common cultivars in North America. Common yarrow is native. Fern leaf yarrow is not, (I think it is from the Caucasus (sp) but has naturalized on the West Coast (and maybe in the Rockies)

Common yarrow (a. millefolium) forms low, (dense, if conditions are right) mats of small, finely cut, dark green leaves that readily spread by rhizomes. In the garden bed, the foliage will form into rounded mounds from which the flower-stems emerge (and the after a couple of years, will sprawl and spread everywhere, including into the yard.) If mowed, the mounds do not form, and the plant does not have the opportunity to send up flower stalks, but will colonize or run along just underground to sprout up in bare or thin places in the grass. It does well in a mixed lawn with plenty of sun. It is drought tolerant. I had a 4'x25' bed of a. millefolium "Colorado Mix" at my old house that readily spread into the surrounding yard, where it got mowed every week. The whites and burgundies were hardier, and spread more readily than any of the other colors. The a. millefolium "Cerise Queen," and some wild white a. millefolium I had transplanted from a field colonized the yard even more successfully. Fern-leaf yarrow - most North American cultivars with those fern-like gray-green leaves and tall, yellow flowers are cultivars of a. filipendulina. (or hybrids with the greek a. taygetea) is a wonderful garden plant, as long as it gets divided every 2-3 years, otherwise, it can get to looking quite squirrelly. It is not at all suited for inclusion in a lawn.

Achillea millifolium is highly variable by region, and it is not clear if different species and subspecies have evolved, or if it is all the same species. If I were going to use it as part of a lawn, I would either transplant wild plants growing in my region, or obtain seed or plants from seed or plants developed in my region. And if it wasn't commonly found growing in pastures, meadows or roadsides in my area, I probably wouldn't expect it to do very well as part of a lawn mix in my yard.