Case first comes across the shuriken while he's walking the Night City streets thinking Wage has a hit out on him. He thinks of how the "shuriken had always fascinated him, steel stars with knife-sharp points" and how "these were the stars under which he voyaged, his destiny spelled out in a constellation of cheap chrome" (1.75). So, when it comes time for him to buy a weapon, he buys the shuriken, obviously. Actually no. Instead, Molly buys it for him after they leave Chiba City as a souvenir.
Then nothing much happens with the shuriken. Case never uses it for any purpose, and it's all but forgotten until the very end of the novel when he finds it weighing down Molly's farewell note. After one last conversation with Wintermute (in the guise of the Finn), Case hurls the shuriken into the wall screen, which "woke, random patterns flickering feebly from side to side, as though it were trying to rid itself of something that caused pain" (24.42).
So why is an object that is only mentioned a handful of times and then literally thrown away an important symbol for the novel, you ask? Well, the shuriken is tied to Case's past. It is a souvenir from Chiba City, where Case spent perhaps the lowest moments of his life. A lowness brought about by events in Case's past that leading him to Chiba City. It is also linked to Case's feelings for Molly, the lover who just left him.
When Case throws the shuriken at the television, think of it as the moment when he finally lets go of his past and the tremendous hold it has on him. He, like the wall screen, is "trying to rid [him]self of something that caused pain."
But it's also worth noting what's literally happening here. Case has just thrown a seriously old school weapon at the heart of a seriously new school tool. But instead of breaking the screen altogether, that old school weapon awakens it. We might think of this as symbolizing change, which is a bit of an obsession with this novel (yep, we've been beating that drum like a dead horse in this learning guide—not to mix metaphors or anything). Change comes no matter what for these characters, and for the most part, that's neither good nor bad. Old gives way to new, and shurikens awaken screens. For better or for worse, Case has survived his past and has a future of hacking to look forward to.
On a side note, Molly's hands contain five-pointed blades as well, one per finger. She literally has a shuriken for a hand. What do you think it says about Molly that she had her weapon integrated into her very body, and is unable to throw it away like Case by the novel's end? Does that mean she'll never be able to let go of her past?