A funnier, less old-fashioned way of calling someone a "wet blanket". Used when referring to anyone who brings down the fun factor of a given situation. Said individual is as much of an annoyance as an relentlessly itchy scrotum.
"Guys, could you use your 30 cm voices? People are trying to sleep"
"Dude, why do you always have to be such an itchy sac about things? Have some fun already."
Shoe itchy is a joke pronounciation of Shuichi Saihara's (from New Danganronpa V3: Killing Harmony) name. Many people make jokes using his name like Pooichi given to him by Miu Iruma.