Connes' critique was recently analyzed by Kanovei, Katz, and Mormann in this article in Foundations of Science (see also arXiv 1211.0244). Here is the abstract:
We examine some of Connes' criticisms of Robinson's infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a 'virtual theory' and a 'chimera', yet acknowledged that his argument relies on the transfer principle. We analyze Connes 'dart-throwing' thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being 'virtual' if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is 'virtual' is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality.
We analyze the philosophical underpinnings of Connes' argument based on Goedel's incompleteness theorem, and detect an apparent circularity in Connes' logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace (featured on the front cover of Connes' magnum opus) and the Hahn-Banach theorem, in Connes' own framework. We also note an inaccuracy in Machover's critique of infinitesimal-based pedagogy.
A brief review of Kanovei-Shelah is here.
Connes wrote in 2001 as follows:
What conclusion can one draw about nonstandard analysis? This means that, since noone will ever be able to name a nonstandard number, the theory remains virtual (Connes et al. 2001, p. 16)
The exact meaning of the verb "to name" used by Connes here is not entirely clear. Connes provided a hint as to its meaning in 2000, in the following terms:
if you are given a non standard number you can canonically produce a subset of the interval which is not Lebesgue measurable. Now we know from logic (from results of Paul Cohen and Solovay) that it will forever be impossible to produce explicitely [sic] a subset of the real numbers, of the interval [0, 1], say, that is not Lebesgue measurable (Connes 2000a, p. 21, 2004, p. 14).
The reference to Solovay indicates that Connes is relying on the result, which may be found in Solovay (1970, p. 3, Theorem 2) on the existence of a model S of set theory ZFC, in which (it is true that) every set of reals definable from a countable sequence of ordinals is Lebesgue measurable. Thus when Connes claims that the theory remains virtual he apparently means that the theory remains undefinable.
This however is contradicted by the existence of a hyperreal field definable in ZF, with transfer probable using only the countable axiom of choice, and a further mild well-orderability assumption (weaker than well-orderability of the reals taken for granted by Connes) ensuring properness; see this answer for details.