no g-d forbid they hang it on the wall
Karaites DO NOT hang their talitot on the wall. They wear talit with white and blue tzitzit, only that the blue dye is vegetable. Since they don't believe in the Oral Law, they reject the idea that Techeiles must come from the chilazon. Furthermore they claim that a shelfish is not Kosher and so cannot be used for a mitzvah. Their reasoning is based in the fact that, when the Temple stood, Jews should not only avoid eating non-kosher but also touching it.
Hanging them in the wall? It's either a myth, or perhaps some really did it in the past (Karaites were not a well defined group in the past, the word applied to anyone who accepted only the Tanach -Mikrah- and rejected Oral Law). No wonder some cult could have done it some time in history, but those we know as Karaites today, certainly do not