You don't need it at all -- it's overkill. The shingles are enough and since the coop isn't heated, you aren't going to get any ice damming. Roofing felt is useful for two things: one, it covers your house FAST in case it rains the night after you strip the old shingles -- it will keep the rain off for a few days. And, two, it gives you nice lines to keep your shingles level. But with a coop that small, you could just snap some lines.
I used some on my 12x8 coop, but it was really rainy when I was roofing -- just did it cause I knew I couldn't beat the rain.