Certainly can't hurt to cut off yellowed or wilted leaves, but I would leave any green ones that look like they could recover. A green leaf is still making food for the rhizomes, so why cut it?
When you dig up a daylily clump to divide and transplant it, you get a huge ball of big, fat rhizomes (well, the size of your fingers, at least) unlike a rose that has all those fine roots. I honestly don't think the roots are damaged that much in transplanting a daylily as long as they don't dry out in the dividing process. That's a challenge on a dry day in June in the deserts of Utah. (the whole place is a desert, just parts of it are irrigated so they don't look like it).
I had a large plastic tray about 7in. deep that I filled with water, and after I dug the clump and hosed off the soil, I'd sit the whole thing in the tub while I messed with dividing to keep the rhizomes moist. I just planted some divisions in my daughter's garden in Salt Lake last month and on this last visit, they are all doing great, and two are blooming!
