They start off by pointing to the similarities between the two elements; similar atomic size, pkA values (which would allow similar acid-base behavior) and electronegativities. These similarities would lead us to believe in the ready replacement of P by As, but that's where they end. From here on the devil is in the details.
Experiments have been conducted with model compounds approximating the phosphate diester backbone in DNA and its putative As counterpart. Firstly the authors note that the phosphodiester backbone is really stable to hydrolysis (the cleavage of the bonds by water), so stable in fact that it's difficult to measure its rate of hydrolysis in DNA because of the slow rate of the reaction. This has led to many studies performed on model compounds where the two linkages to sugars in the DNA backbone have been replaced by suitable alkyl groups. These studies have measured the half-life of the phosphate diester linkage. The half-life of a reaction measures the time taken for half the reaction to finish and is a very convenient tool for quantifying its rate; in case of the phosphate diester compounds, it turns out to be a whopping 30,000,000 years. This is a huge achievement if you consider the very high concentration of water in cells which is 55 M. As the authors say, this means that only two out of the 3 billion base pairs in the DNA from a human cell are expected to undergo spontaneous, uncatalyzed hydrolysis per week. It is a testament to the amazing power of catalytic enzymes that this reaction is brought within reasonable time frames in biological systems. While the results from the model system constitute an extrapolation to DNA, it's a reasonable one since the accessibility of the P to attack by water in the model compounds is similar to that in DNA.
The corresponding arsenic diesters present a scenario that's out of the ballpark. The half-life of the model arsenate diesters is no more than 0.06 seconds which corresponds to a difference of a factor of between the two. This is an absurdly large number; as just one comparison, it exceeds the number of cells in the typical human body by a factor of ten thousand. Error bars will do nothing to change it. In fact the hydrolysis of arsenate diesters is so fast that this fact has been used productively by scientists who want to study the kinetics of phosphate containing molecules but who are thwarted by the extremely slow nature of the reaction; substitute the P with As and you get a system which will otherwise be similar but which will transform itself rapidly into the desired products. What biology abhors, chemists can adore.
The magnitude of the problem is driven home by the calculation that with this rate of hydrolysis, half of all the arsenodiester linkages in the DNA of the GFAJ-1 bacterium would be cleaved in less than a tenth of a second. In addition there are other problems with As. As the authors note, As can also change its oxidation states more easily compared to P. The oxidation state of As and P in the DNA backbone is +5. But unlike P, As can undergo ready enzymatic conversion to a +3 oxidation state. Compounds containing As +3 are even more unstable than those containing As +5.
Now does this mean that organisms substituting As for P cannot exist? No. But as basic chemistry demonstrates, this would present very great challenges of stability. As indicated, there could be possible solutions to this problem such as dehydrating conditions containing very little water or the presence of special proteins that stabilize DNA and shield it from water. But it's clear that any such extremely novel ideas would be speculative at best until supported by evidence.
I hate quoting Carl Sagan all the time but his statement about extraordinary claims requiring extraordinary evidence is a cliche because it's true.
Fekry, M., Tipton, P., & Gates, K. (2011). Kinetic Consequences of Replacing the Internucleotide Phosphorus Atoms in DNA with Arsenic ACS Chemical Biology DOI: 10.1021/cb2000023