StringIOでエラーが出てて辛かったけど，BytesIOがあるらしく，それを使うとできました．
Python3とPython2で文字列の扱いが異なる仕様がファイル読み込みまで影響が出てるんですね．
import io
import urllib.request
from PIL import Image
f = io.BytesIO(urllib.request.urlopen("画像のURL").read())
img = Image.open(f)
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