It is an eminently stupid thing to do, which is why I don't think he's doing it at the moment. Come what may, he's got the job until the end of the season. After that, the summer will arrive, bringing with it a whole host of famous names with glittering CVs just waiting for a good offer. We have already established that Levy holds grudges for a long, long time: I still don't think he's forgiven United for their pursuit of Berbatov in 2008, for example. What is the point, then, of antagonizing Levy (who isn't a man I'd imagine is swayed by impetuous demands by his manager: see Redknapp's 'new contract' for details)? All a call of 'back me or sack me' will do is make Levy bear a grudge that will only make it harder for Tim to stay on come May, even if he's done a good job, and given his lack of experience I don't think a club of our size would take him on should we decide to let him go. It's career damaging to demand anything of the chairman at this point, especially when he's already given you the job temporarily despite a public outcry and the fans' wariness over your suitability for the role.
Tim has nothing to gain from making a 'back me or sack me' claim, imo. He's already got the job till May, and it's not like he's beloved by the squad or adored by the fans: in fact, the person who most ardently believes in him is probably Levy, judging by the way he's kept Tim on the same 'transfer committee' that big names like AVB and Baldini have been on. So no one will unduly miss him should he decide to leave: he hasn't built up enough leverage yet. Ergo, it's a stupid move, and I don't think Tim is dumb enough to make said move.
No, this seems legitimate enough, which is worrying. My worry is that Tim is undermining Baldini via his close links to Levy, and my hope is that Levy has the balls and the brains to stand firm in his decision to appoint a DoF again: otherwise, we will have wasted the biggest windfall in our history and taken several large steps backwards, all while becoming the same old laughing stock we were pre-2010.