Your question is fascinating, and it resonates with our very real obligation to express our gratitude to Hashem for the miraculous survival of Israel and to show our solidarity with our people in their distress.
Interestingly, there is a tradition that the Baal Shem Tov cautioned against iron covers because of iron’s association with war, and consequently some communities have a custom not to use iron and steel covers. This custom is based on the Torah’s prohibition against sculpting the stones of the Holy Altar (that “prolong life”) with metal instruments (that “cut it short”).[1] Others avoid covers made of any metal.[2]
However, common practice follows the classical authorities who do not apply this analogy to mezuzah covers and allow all materials, including iron.[3] Thus, unless one is a member of those communities who are strict in this matter, there would be no objective halachic prohibition against using the missile scrap.
But here’s where the issue becomes subjective and nuanced. On the one hand, one could suggest that even according to the lenient approach, perhaps making a cover specifically out of a weapon of war is not appropriate. Yet, on the other hand, if the cover is meant to celebrate Hashem’s protection by accentuating the defensive aspect of the iron dome, perhaps it is not to be taken as a symbol of aggression, but of Divine providence. In other words, “Is it a sword or a shield?”
There is a further subtlety as well. Does the symbol indeed communicate that Hashem is the protector of Israel, or does it mean to say that Israeli technology and Jewish brains are our protector? Indeed, one online seller advertises:
Protect your home with mezuzot from Sederot! Own a handmade mezuzah cover, made from fragments of the Iron Dome that protected and continues to protect Sederot and all the People of Israel!
Bottom line, I am not aware of any authority who allows iron and steel covers yet forbids “Iron Dome” fragments. Considering the ambiguity of the symbol, I would suggest that a person who purchases one should be clear that his intention is to express his gratitude for Hashem’s miracles and his solidarity with the People of Israel.
[1] Da’as Kedoshim 289:1 brings this tradition but limits it to iron alone.
[2] Sha’arei HaMezuzah 16:2.
[3] R. Ovadiah of Bartenura, Keilim 16:7; Maharil Chadashos 122; Aruch HaShulchan 286:5; Yafeh L’leiv 3:289:1.