I've never heart of it actually killing a daylily but if it kills a lot of leaves it could theoretically weaken them if they're not particularly robust. Like daylily rust, cultivars vary in their resistance/susceptibility to leaf streak. It is said to need an injury, such as pest or weather damage, to infect a plant so differs from daylily rust in that respect.
No, cold winter climates don't kill it unfortunately. What kills rust in cold winter climates is that the leaves die back and the mycelium (the body of the fungus) in the leaf cannot survive winter without living foliage. It can, however, use the "winter spores" to infect the alternate host, patrinia, in the spring where that plant is present. The "winter spores" of daylily rust do survive very severe winter climates (it was first recorded in Siberia) but can't re-infect daylilies, only patrinia, and then return to daylilies again.