Let's run through this nice and quick.
The theory goes that there's a white tennis shoe hanging from a tree in the jungle, as we saw in the first few minutes of Lost in the Pilot episode.
Later, days later, Christian Shephard appears to Jack on the Island. Bizarrely, in the episode White Rabbit, he appears to be wearing white tennis shoes!
Therefore comes the theory that Christian Shephard is alive and is physically on the Island and physically wearing the shoes we saw one of hanging in the tree. Never mind the fact that he appeared here. . .
. . . out of nowhere, and disappeared just as quickly into thin air. (Notice how we can't see his feet.) Never mind we only saw one tennis shoe and yet Christian appears to be sporting a pair on the Island. Just so we're clear, he doesn't always wear tennis shoes.
White tennis shoes would appear to be an odd thing for him to have on his feet to be buried in, right? So, for whatever reason, the real, living Christian took the tennis shoes he found in the jungle and that's what he wears, right? So the theory gathers dust for a couple of years, and then along comes the episode The Beginning Of The End. We see Christian in Jacob's cabin. Most notably, we see his feet.
Look! What is clearly (well, not that clearly) a white tennis shoe there on Christian's foot! Suddenly the fans of the Christian-is-really-alive have EVIDENCE that their theory is right!
May I point you towards the mobisode So It Begins. This was a scene that took place moments before the first ever episode of Lost began. It shows Vincent running through the jungle and encountering Christian Shephard. Brilliantly, we get to see Christian's feet.
He's wearing a pair! So therefore the white tennis shoe we see hanging in the tree just minutes from this moment cannot possibly be the same as what Christian has on his feet. The physical evidence is killed. The theory void. I can't explain exactly what is going on with Christian Shephard's appearances on the Island, but at least we can lay the matter of whether he's wearing the tennis shoe to rest.