To answer the title question, in an omnibus spending bill if you give enough people a reason to vote against it, by going over their individual "red lines",
that can sink a bill.
In this case, the Republicans had basically announced that they would vote against it, along the lines of:
A notice from House GOP leadership to Republican lawmakers states that "Republicans will not aid the Democrats in their socialist tax and spending spree."
So any further "defections" from the Democratic camp on the bill (over whatever reasons) could sink it, because they have a narrow majority. I haven't followed all the special elections, but back in April, this kind of situation was essentially predicted:
the narrow majority means that essentially any two [Democratic House] members can have a veto over certain legislation. Progressives in the House may also try to use their leverage to block legislation which they believe does not go far enough.
Also, the affair provided the Republicans the opportunity to accuse the Democrats of caving in to anti-Semitic elements in their midst:
Republicans, who refused to support the bill themselves, even with the Iron Dome provision included, quickly accused Democrats of undermining Israel’s security.
House Minority leader Kevin McCarthy tweeted: “Democrats just pulled funding from the Iron Dome — the missile defense system that has saved countless lives in Israel from Hamas’ rocket attacks. While Dems capitulate to the antisemitic influence of their radical members, Republicans will always stand with Israel.”
Regarding the separate legislation on the Iron Dome, which was quickly introduced, at least Rashida Tlaib had announced she was going to vote against it. Ilhan Omar also expressed disapproval of this latter bill. Most recent news is that
The measure cleared the House in a bipartisan vote of 420 to 9. Eight Democrats and one Republican voted against the funding for the missile defense system. Two other Democrats — Hank Johnson of Georgia and Alexandria Ocasio-Cortez of New York — voted "present."
There's a full list now available of who voted against:
The "no" votes were Reps. Ilhan Omar, D-Minn.; Rashida Tlaib, D-Mich.; Ayanna Pressley, D-Mass.; Cori Bush, D-Mo.; André Carson, D-Ind.; Marie Newman, D-Ill., Raúl Grijalva, D-Ariz.; Chuy Garcia, D-Ill.; and Thomas Massie, R-Ky.