How serious a problem is this for LQG? Does this mean that LQG doesn't have GR as its semiclassical limit? Does that mean it's a dead theory, or maybe just that it needs to be modified? Is the technique using Euclidean gravity reliable?

Since I'm not a specialist, I'd be interested in a hand-wavy explanation of what the Euclidean gravity technique is about.

A few months later Ghosh and Perez posted an LQG paper http://arxiv.org/pdf/1210.2252.pdf
that references Sen (their reference [19]) At the top of the second column on page 5, the paragraph right before the Conclusions section, they say there is no reason for the two sets of log-corrections to be mathematically identical since they arise from different origins.
One analysis uses entanglement entropy, the other does not. However they say that the two approaches should be COMPATIBLE (not sure what that means) and they plan to do some calculations to investigate this. I can't give you an answer, just offer this reference.
==quote Ghosh Perez==
Coming back to our expression (18), we have seen that the details of the scaling of the BH entropy depend on the renormalization of the various couplings including in particular G(l). Such renormalization should take place at length scales where matter loop effects are important. Furthermore, matter loops may produce other sub-leading corrections to (14), such as log l. These corrections give rise to logarithmic corrections log A to the entropy (18). In the context of entanglement entropy log-corrections have been obtained by several authors [17– 19]. These log-corrections should be compared with the log-corrections to g_{M}—in fact, it will be an important benchmark test of the semi-classical limit of LQG. Notice that these log-corrections are in general very different in origin from the so-called LQG log-corrections to (18). So it is not surprising that these corrections do not agree [19]—there is no reason for these to agree. However, it is crucial that the log-corrections to g_{M} and those in the entanglement entropy are compatible with each other. We intend to carry out these calculations in future. LQG also suggests that the matter contribution to entanglement entropy coming from sum over all loops is finite!

Mitchell Porter pointed out on StackEx that LQG now has several versions of BH entropy. Stay tuned

If you want, for comparison sake, the LATEST LQG paper on BH entropy, it happens to be on the Poll for second quarter 2013.
Black hole entropy from KMS-states of quantum isolated horizons

Daniele Pranzetti
(Submitted on 29 May 2013)
By reintroducing Lorentz invariance via a complex connection formulation in canonical loop quantum gravity, we define a geometrical notion of temperature for quantum isolated horizons. Upon imposition of the reality conditions in the form of the linear simplicity constraints for an imaginary Barbero-Immirzi parameter, the exact formula for the temperature can be derived by demanding that the horizon state satisfying the boundary conditions be a KMS-state. In this way, our analysis reveals the connection between the passage to the Ashtekar self-dual variables and the thermality of the horizon. The horizon equilibrium state can then be used to compute both the von Neumann and the Boltzmann entropies. By means of a natural cut-off introduced by the topological theory on the boundary, we show that the two provide the same finite answer which allows us to recover the Bekenstein-Hawking formula in the semi-classical limit. The connection with Connes-Rovelli thermal time proposal for a general relativistic statistical mechanics is worked out.
Comments: 10 pages, 1 figure http://arxiv.org/abs/1305.6714

Pranzetti recently gave a talk at Perimeter which is online as video. One of the authors of the first paper I mentioned, Alejandro Perez, is an invited speaker at Loops 2013, at Perimeter this month and his talk will most likely also be online video. So if Ashoke Sen's paper is important Perez will most likely be asked about it in questions after, or may refer to it in talk. I'll be interested to see if it comes up. Perez cited the Sen paper as reference [19], so I guess there is a chance that it will come up in the main body of his plenary talk 3 weeks from now. We'll see.

My view is that Sen's result says nothing about LQG's viability, but suggests that the calculations of black hole entropy in LQG are wrong, ie. what's calculated is not the sppropriate qusntity, something which can be guessed from the LQG literature itself).

I believe the indications from string theory are that something like LQG is a theory of quantum gravity, but without classical GR or fine classical geometry. Or at least that's my reading of Castro and colleague's "The gravity dual of the Ising model", which like LQG is pure gravity without unification. The other piece of evidence is Brian Swingle's AdS/MERA which applies an LQG like formalism to the Ising model to yield something like a coarse geometry. Even Maldacena lately wrote about very quantum geometry, whatever that is. The interesting thing is that string theory makes suggestions about what ingredients to add to go from coarse geometry to fine classical geometry.

Also, even in cases where the entropy is A/4 in string theory, it appears, following Ryu and Takayanagi, that the object is not necessarily a black hole. Bianchi (LQG theorist) and Myers (string theorist) have a speculative paper following up on this. http://arxiv.org/abs/1304.4926 (A/4 for non black holes) http://arxiv.org/abs/1212.5183 (Bianchi and Myers)

Since I don't know any of the details, I am probably missing something elementary, but why is this a problem for LQG? It seems to me that this means Euclidean gravity and LQG are different, but could still have the same classical limit. The semiclassical limits may differ, they could be slightly different approximations to the correct one.

In LQG and quantum gravity, semiclassical gravity usually means classical GR plus quantum matter fields. LQG is a theory without matter, so the semiclassical limit is classical GR, eg. http://arxiv.org/abs/1110.5899.

OTOH, Sen attributes the missing term to the some contribution from the "massless graviton loop", which he believes is reliable, even without a full theory of quantum gravity. I don't understand his calculation well enough, but he refers to http://arxiv.org/abs/1104.3712 which has a section "Entanglement entropy as one-loop quantum correction", which says "A natural point of view on the entanglement entropy of black hole is that this entropy, as was suggested by Callan and Wilczek, is the first quantum correction to the Bekenstein-Hawking entropy. Indeed, the Bekenstein-Hawking entropy SBH can be considered as classical, or tree level, entropy." This is similar to other cases such as http://arxiv.org/abs/gr-qc/0207118 and http://arxiv.org/abs/hep-th/0211072 in that some contribution from the one loop terms include the first quantum corrections and are believed to be reliable. Since these include the first quantum correction, they can be considered "semiclassical".

Sticking with the latter definition of "semiclassical", since these first quantum corrections in GR treated as a quantum effective field theory are believed to be reliable, it is expected that LQG should reproduce them if it reduces to a theory with GR as its classical limit. The reason is that we already use the reliability and smallness of those first quantum corrections to say that classical GR is a great approximation to quantum GR. And quantum GR is the theory we really use, since everything is quantum.

Most calculations involving the entropy of black holes involve horizon physics. The horizon is a very peaceful place for large blackholes (the near horizon geometry is simply given by Rindler space).

Calculations for the entropy are thus given to fantastic precision by the use of effective field theory techniques. Semi classical gravity is the approximation where you fix a background metric, and allow matter to be quantum mechanical to all orders, and expand the effective action to one loop only in the gravitational field (so you are always intrinsically talking about gravitons). In this sense, its almost quantum gravity, but not quite, b/c you are throwing out higher derivative terms, nonperturbative terms as well as ignoring backreaction.

However, b/c Newtons constant is so tiny, and the horizon essentially has no curvature, this theory is very nearly exact. So even though you have only calculated to one loop precision, further refinement in the approximations will only involve tiny corrections. This is the reason why Hawkings entropy calculations as well as the numerical prefactors are expected to be universal for any more refined microscopic theory of quantum gravity and why these Euclidean calculations are believed to be nearly exact.

As far as LQG. I don't know what the status is on their calculations, but I believe they don't really have agreement on how to do the calculation in general. Further the Immirzi paramater can always be tuned to agree with one type of bh solution (but then you have to get the rest right).

You may have noticed that the second day of Loops 2013 is focused mainly on BH entropy and the two plenary speakers (Perez and Bianchi) have both found S ≈ A/4 independent of Immirzi.
So the organizers of Loops 2013 seem to have made it "official" that in LQG the entropy does not depend on the Immirzi, to leading order.

We'll have to see what Alejandro Perez says, in his talk, about Sen's work. As I mentioned earlier in post #2 of this thread, he refers to it in a recent paper and explains some difference in a second order or log term.

A good way to catch up on the status would be to watch the video of Perez' talk next week. Including the questions afterwards, I would guess. That will be Tuesday morning. Abstracts of the BH talks in the afternoon (parallel session) indicate they are tending to use a variety of approaches, as you say, but AFAICS they would all be A/4 to first order and thus not "tunable" in the old way, which used a real Barbero-Immirzi parameter. I could be wrong about some of the afternoon talks. But the best indication of current status will be the plenary talks by Perez and Bianchi in the morning.