I'm going to answer a slightly different question about consequences of a 125 GeV Higgs for low-energy supersymmetry. For internal consistency, supersymmetry seems to be a requirement at very high energies (or perhaps just on the worldsheet) in string theory. So these questions are indirectly (and tenuously) connected.
This also seems to be the approach Luboš Motl took in his answer, but I think he conflates some issues that should be disentangled. The bottom line, in my opinion, is: 125 GeV is an unexpectedly large mass in the MSSM. If supersymmetry at the TeV scale is correct, this Higgs mass strongly hints at one of three things: either very heavy scalar superpartners, out of reach of collider searches (though fermionic ones may be lighter), or a SUSY spectrum with very large mixing among scalar top quarks, or an extension of the minimal model with new interactions for the Higgs boson. Only the last of the three can evade the conclusion of fine-tuning, though one can debate whether Nature cares about what we call tuning.
One typically hears that supersymmetry requires a Higgs mass below 135 GeV. For instance, you can find such a statement in equation 45 of a review article by Carena and Haber (which can also serve as a source for other statements I'll be making in this answer; I won't try to be exhaustive with references to the original literature). Let's unpack that claim a little, which is also related to this other recent question about the little hierarchy problem:
The MSSM: This is the minimal supersymmetric Standard Model. In this model, the Higgs has gauge interactions and the Yukawa interactions it needs to give mass to SM fields, and no other interactions. This is very predictive. At leading order, it predicts $m_h < m_Z$ = 91.1876 GeV. So the MSSM is always in some tension with larger Higgs masses. There are corrections to this Higgs mass formula, however, from quantum corrections arising from its interactions with supersymmetric partners of the top quark (scalar tops or "stops"); roughly, there are corrections going as $m_t^4/v^2 \log m_{\tilde t}^2$ with $m_{\tilde t}$ the stop mass, and corrections going as $m_t^4/v^2~X_t^2/m_{\tilde t}^2$ and $m_t^4/v^2~X_t^4/m_{\tilde t}^4$ where $X_t$ is a measure of mixing between left- and right-handed stops.
So, in the MSSM, a large Higgs mass requires large stop masses. The masses required are a little less large when the stops are highly mixed. Now, the number 135 GeV for the maximal allowed Higgs mass is derived assuming stops are below 2 TeV, although I think a more modern version of the calculation would conclude that even 135 is out of reach. On the other hand, dropping the assumption of stops below 2 TeV, in the MSSM with extremely heavy superpartners the Higgs mass could even be a little above 140 GeV.
The MSSM, with large Higgs masses, is tuned: both the stop mass and the mixing $X_t$ show up in quantum corrections that want to shift the electroweak breaking vacuum. Most people working on the MSSM have studied models with light superpartners, below 1 TeV, so that this tuning is relatively small. On the other hand, some have studied "split" models with heavy scalars, giving up on solving the fine-tuning issue. A 125 GeV Higgs is a better fit to those scenarios, with scalars in the ~ 10 TeV regime, than to "standard" MSSM models. It is still compatible with more standard MSSM scenarios in the limit of large stop mixing, however (though this imposes strong and interesting constraints on the model!).
Luboš is advocating the split models, and they are interesting; nature may not care what we consider to be "fine-tuned." Also, one sometimes encounters claims that the fine-tuning is reduced in some versions of these models. Trying to quantify precisely what is meant by tuning is a can of worms I'll avoid opening in this response. I don't agree that these slightly split models with 10 TeV scalars are more "stringy" than "QFT-like," since all of the calculations are done in effective supergravity theories, although they do lend themselves to solving the moduli problem that arises in string theories (see this and this, as well as the more recent work of Gordy Kane that Luboš cites).
Beyond the MSSM: Supersymmetric models can also accommodate a Higgs with more than the minimal set of interactions. In that case, new contributions to its mass can arise already at the leading order, and the tension with fine-tuning described above is much smaller. There are too many conceivable versions of this to classify in this answer, and their implications for colliders can depend on details. But it's certainly interesting to consider models without tuning and with new physics beyond the MSSM.
Bottom line: It's still too early to say definitively if the hints at the LHC are evidence of a 125 GeV Higgs. If so, then the next few years--possibly even 2012--could tell us if we are in one of these three scenarios (split MSSM, MSSM with highly mixed stops, beyond the MSSM) or if supersymmetry is completely absent at the weak scale.
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