It seems to me that the teams who would value Posey's services are those teams who already have a solid core of star veterans. And among those teams, only those teams who need a swingman SF/SG. Posey is not at the level where he should be the second-best, or third-best, player on a playoff bound team. So which teams fit those criteria? To my mind, the two most obvious are Boston and LA.
Both are over the cap. However, LA has a slight advantage if Posey is simply looking for more money. LA's core is younger, therefore, they may be willing to sign him up to a longer contract because their contending window, theoretically, is longer. Another thing is that Posey and the return of Bynum, theoretically, would complete the Laker puzzle. They can let Turiaf go and not miss him. They'd miss Vujacic, but Farmar should be adequate.
The problem for Boston is that keeping Posey won't complete their puzzle. They can use the MLE on Posey, sign him up for multiples years, and they'd still have a hole to fill at the backup Center. Heck, you could argue that they'd still need a backup PG! Having said that, I am leaning towards offering the full MLE to Posey. And then just sucking it up and paying up some more luxury tax to get a backup Center. The Celts just may have to make do with Pruitt at the backup PG position, along with Eddie House if he chooses to come back for the min.
If the Celtics lose Posey, and the Lakers let Vujacic go. I would encourage Danny to go after Vujacic. Vujacic can play backup PG, SG, and some SF. Maybe Giddens can get some small time at SF and play competent defense. I don't think that would be a bad backup situation, with Powe, House, Vujacic, and a veteran big man (even Pollard would work fine if he can stay healthy). The Celtics would have a really outstanding three-point shooting bench with House and Vujacic.