I would advise that everyone take more care when assessing the Turing Completeness various models of computation. Many analyses that I have seen do a check for logical completeness, but forget to rigorously check that you have scalable and fully accessible memory, even in the most generalized cases. Even if you assume that some resource is unbounded, your other mechanics might not actually allow you to interface with it, even if they do work on a constant/small scale (locality of computation does not necessarily imply this). You could take a turing machine its infinite tape, but forget to show that you can move it left or right an arbitrary number of times. On the flip side, another ridiculous assumption involves including the power of human operation. Yes, an abacus is "Turing Complete" if it is infinitely large and you, as a human being, do the logic in your head. Pen and paper is Turing Complete by the same reasoning (this is ridiculous). Yes, we know that humans are capable of computation. I might grant that you have scalable memory in these situations, but where's the automation of your logic?

Here's an example of the first: I suspect that the in-game mechanics of a generalized version of Minecraft to not be Turing Complete. By generalized, I mean that you remove the chunk loading and update limitations, giving you a countably large playing field. I, of course, am not considering limitations (or benefits) of Java, the machine running it, or unintended behavior (bizarre bugs). Forgo things like command blocks and bizarre uses of spawn eggs/entities, just consider legitimately obtainable resources (we can assess TC with these later).

To briefly sketch the idea, when you have a constant sized machine (accessing arbitrarily sized input), the selected mechanics are too limited by construction to "grow" the equivalent to a Turing Machine's work tape. A superficial reason for this is because the mechanics are coded to only extend locally (note that this is distinct from the "computation is local" result). For example, using pistons to push only goes out 11 blocks before it fails. You can chain multiple of these together, but only a constant number. You also need controllable Redstone signal to extend far enough to change their states. With a bit more work, you can show that mechanical self propulsion using pistons is not possible (there's a giant thread on Minecraft Forums concerning this), even in one direction. Three simple reasons for this: 1. You cannot push extended pistons, 2. you cannot push/pull/power your back most piston without "leaking" blocks, and 3. you cannot make a pushable clock. Your ability to shift around and access your tape/memory or move your Turing "head"/logic in this approach goes out the window.

For block based memory (to make use of infinite "block space") in general, you not only have to move a number of blocks of an arbitrary number to locations an arbitrary distance away, but you have to autonomously generate and destroy an arbitrary number of blocks, too. Cobblestone, Ice, Pumpkins, and Melons all do this and are thankfully Redstone distinguishable, so we at least have our endless supply of symbols for such a tape. There are additional requirements and engineering challenges, but they fall outside the purpose of this post.

Issues like these are not considered in almost all the claims Minecraft's Turing Completeness that I have seen. They absolutely should be. Obviously there may be ways around this, but I hope my point is clear. Make sure the application of your "proofs" or constructions actually make sense. Make sure they demonstrate how to handle the memory and the logic (algebra) necessary. And for god's sake, make sure that your model is actually an automaton (I'm looking at you Magic and LBP).

On a side note, I did start compiling a paper on the computational complexity of Minecraft a while ago, but left it unfinished. It goes through various generalizations, starting from some most basic (just blocks, redstone, etc.), adding some components like randomness, commandblocks, entities. Even some structural assumptions such as superflat worlds and tiled machines (works for Conways game of life). I lost some motivation because I did not believe people would care enough to be worth my time. Even if I were to pick it up again, the game has changed quite a bit since 1.7 came out. My results might be outdated (new commandblock functionality changes things greatly). However, if others show interest, I could perhaps start working on it again.