The same thing that happens with most economic boycotts: not much. From the Weekly:
The city continues to do business as usual with Arizona companies. Aside from a single conference in Phoenix that LAPD officers were forbidden to attend, absolutely nothing has changed. "It's easy to engage in the political theater of boycotts," says Chris Newman, general counsel of the National Day Laborer Organizing Network. "It becomes very difficult when you start talking about dollars and cents."
In a few weeks, the City Council will be asked to sign off on a $90 million sewage plant contract with Honeywell, and you can expect lots of push-back from immigrant activists, led by Councilman Ed Reyes. The problem is that Honeywell isn't even based in Arizona - the company is headquartered in Morristown, N.J. The division that will be handling the work is in Minnesota, and the parts are manufactured in Texas. If Honeywell doesn't get the business, it could mean the loss of 80 L.A. jobs. None of which sounds like a huge loss for Arizona.