This was not only envisaged in the original 1947 UN partition plan (under which Jerusalem would become a corpus separatum under international control), but was also discussed with regard to the Jerusalem holy places (although not the broader city) in final status negotiations in 2000-01 and 2007-08. My own personal favourite was the serious proposal to declare the Haram al-Sharif/Temple Mount under the sovereignty of God, with a "caretaker" international committee to administer it in his/her absence. Clever politics, that.
The Saudis certainly don't see the holy cities as a "burden," and would regard any effort to end their control as a fundamental, and indeed near-existential, security threat. I'm also not clear on how this would promote reform in Saudi Arabia (I can actually see it strengthening Wahhabi Salafism within the kingdom), or the broader Muslim world. It is hard to imagine Sunnis as a whole agreeing to any Shi'ite role in administering the cities—its rather like proposing that the Vatican share St. Peter's with the Mormons.
On the challenges of multi-religious administration of a holy site, it is both informative and amusing to look at the problems associated in the sharing of the Church of the Holy Sepulchre and the Church of the Nativity between Christian denominations. Squabbles at the former regularly involve the Muslim caretakers or Israeli police being called out to separate battling monks, while disputes at the latter helped spark the Crimean War.
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