Broadly speaking, the most significant ramification of the 2003 invasion of Iraq was the fermentation of conditions of instability in the region. The invasion turned Iraq from a major regional power with great ability to act on the region into a weak state mired in internal conflict and heavily acted upon by other states such as Iran and Saudi Arabia and their proxies. It resulted in a 2006 civil war, left a weak and unstable democratic government, and created a power vacuum filled by various militias representing different religious and ethnic groups. This includes the Islamic State/Daesh which began as an al-Qaeda branch until it emerged as its own group, taking advantage of the new Iraqi government’s weak hold on its territory.
In this context, Iraq’s fall has been to Iran’s gain. The 2003 invasion of Iraq removed it as a state capable of challenging US hegemony in the region. However, without a strong Iraqi state to act as a buffer to Iran, Iran has been able spread its regional influence and pose a greater threat to US interests, both within Iraq and beyond it in the Levant and Gulf. It has done this both by appealing to Shiites in foreign (often Sunni-led) states or funding Shiite militia groups such as Hezbollah in Lebanon and Syria or Houthi rebels in Yemen. This has led the Sunni-led monarchies (excluding Qatar) in the GCC worry about the security threat of an Iran-backed “Shia crescent” surrounding it on its North, East, and Southern frontier.
As for my obligatory tie-in to Israel, the US invasion of Iraq has been mostly advantageous for Israeli interests. Under Saddam Hussein, Iraq posed a major military threat to Israel, with his regime gone so too is the threat Iraq posed to Israel. Israel has also been a key backer of Kurdish independence groups in Iraq (Fache 2019).
With the fall of Saddam Hussein as a pan-Arabist (and therefore fiercely anti-Israel) leader coupled with the rise of Iran, Israel has benefitted from improved relations with Arab states, especially in the Gulf mostly out of shared strategic opposition to Iran.
This is not to say that the destabilization caused by the 2003 invasion has not presented some issues for Israel. While it is true that Israel no longer has to worry about Iraqi scud missiles or chemical weapons landing on Tel-Aviv, Israel does now have to worry about the increasing influence of Hezbollah and Iran in Lebanon and Syria. With the collapse of central state authority in Iraq after the 2003 invasion, Iran has been able to recruit Iraqi Shiite militias to support its aims in the Syrian theatre. Israel has launched numerous strikes in Syria to prevent Iran and its Shiite militias from arming Hezbollah or otherwise threatening its borders (Al-khalidi 2021). One could even pin Israel’s brief skirmish with IS militants in the Golan heights back in 2016 (Osborne 2016) to the power vacuum left by the US in Iraq from which IS/Daesh emerged.