The line item veto is, put simply, a Bad Thing.
Now then, you might ask why, when you consider that it allows the executive, whether it be the President or a state governor, to allow parts of a bill to pass through without having to just accept other parts that would have derailed it, such as a poison pill clause?
The thing is, that's the whole point. One of the great checks on both the executive branch and the legislative branch in the standard three branch system we use in the US is that the legislature sends bills to be approved by the executive as a whole. It acts as a check on the executive by forcing him or her to weigh the fallout of vetoing a bill that has parts that they may disagree with, but is popular overall. (For a good example of that, see the circus around the National Defense Authorization Act in 2012.) Conversely, it acts to check the legislature by forcing them to weigh the fallout of putting a clause in a bill that will guarantee that the executive will kill the bill dead. Thus, both branches are forced to play nice, or else it's going to look bad for the both of them.
The line item veto short-circuits that. With the line item veto, the executive can just excise portions of the bill that they don't like, which is in of itself a gross usurpation of legislative power by the executive. Worse, it means that the executive and legislature no longer have to work together - the legislature can toss in all sorts of horrible things, knowing full well that the executive will strike them down without stopping the bill. The check on the executive and legislative branches disappears.
The fact that the line item veto is popular highlights the public's desire to see the government made to work, even if it breaks the checks that are supposed to keep the place in line. Which isn't all that healthy a desire, to be honest.