Once a mathematical theorem is proven true like the Halting problem can it ever be disproven?

A theorem, once (correctly) proven, cannot be disproven. That said, there are two qualifiers here.

• The theorem's proof must be genuinely correct. But proofs can be quite complicated, and mistakes in them can be very subtle. See this MathOverflow question for a number of examples of theorems which were widely believed to be proven but later shown to be false. This is not likely to be the case with the unsolvability of the Halting Problem, the proof of which is quite simple.
• The theorem must be correctly stated. In particular, theorems are often summarized inexactly for general use; in this case, the inexact but popular summary of the Halting Problem is "no computer program can detect whether or not a given computer program will halt on a given input". But this is an incorrect statement of the theorem, which relies on the Church-Turing thesis - which states, essentially, that anything a person would call a "computer" is fundamentally equivalent to a Turing machine. The article you read suggests that quantum computers do not abide by the Church-Turing thesis, and are not equivalent to Turing machines - the unsolvability of the Halting Problem isn't incorrect, it just doesn't apply to those computers.

As a side note: The standard proof that the Halting Problem is unsolvable is very flexible, and could likely be modified to apply to anything that functions remotely like a Turing machine. The reasonable conclusion here is that a quantum computer is simply not remotely like a Turing machine.