The Economist explains why Israel is so afraid of a nuclear-armed Iran:
Israeli officials say a nuclear-armed Iran would lead to “cataclysmic” changes in the Middle East. America would be weakened and Iran become dominant; pro-Western regimes would become embattled, and radical armed groups such as Hizbullah in Lebanon and Hamas in Gaza would feel emboldened.
Saudi Arabia, Egypt and others could, in turn, seek their own nuclear arms. In a multi-nuclear Middle East, Israel’s nuclear arms may not ensure a stabilising, cold-war-style deterrent. “If Iran gets nuclear weapons, the Middle East will look like hell,” says one senior Israeli official. “I cannot imagine that we can live with a nuclear Iran.” For Israel, 2010 is the year of decision. Yet its ability to destroy the nuclear sites is questionable, and such a strike may precipitate a regional war, or worse.
That’s the best explanation I’ve heard: if Iran was allowed to develop nuclear weapons, or was getting worryingly close, Arab states would implement crash-course nuclear programs. It would set off an arms race across the region, and tensions would rise with it.
In addition to the arms race, and the elimination of Israel’s nuclear advantage which protects them from any serious attack, while Iran is the only nuclear power in the region besides Israel, Iran would be free to fully use their connection with the Shi’a in southern Iraq, and convert it into an actual sphere of influence or, worse, take full control altogether. This would give Iran undeniable power in the region — they would have nuclear weapons, control Hezbollah and have strong influence over Hamas, control a large portion of the region’s oil, and be right on the doorstep of Saudi Arabia’s oil fields.
These two factors could completely destabilize the Middle East. Iran would be, at that point, the uncontested dominant power in the region.