I imagine that the remaining Arab countries will band together against the threat of an Iranian dominated Middle East and they will probably tone down the anti-Israeli rhetoric since Iran is a common enemy of both the Arab states and Israel.
Expect Iraq itself to be very unstable. The Arab world will fund Sunni resistance against the fundamentalist Sh'ia regime and the resistance will probably be under the banner of remaining Ba'athist elements. Neither the USSR, nor the US were friends with Tehran so expect them to support resistance against the Shi'a regime too. This way you could get a civil war between the Arab Socialist/Nationalist, secular Sunnis under Ba'ath leadership and the Iranian imposed puppet regime.
Btw, many Iraqi Shi'ites themselves were anti-Iranian themselves so the Shi'a regime will have to deal with dissent within their own community as well...