Take a look at Gil's post on the idea of using "teiku" on questions of faith. He attempts to show that not only do chareidi rabbis like R. Matisyahu Salomon use it when running up against apparent contradictions and difficulties in the realm of faith, but so did R. Soloveitchik. Therefore, he concludes, critics of R. Salomon are being disingenuous.
Back when the controversy about R. Salomon broke out, there were indeed bloggers who criticized R. Salomon for using the idea of teiku as an ideological cop-out, to avoid dealing with Torah and science issues. In other words, they saw what R. Salomon was doing and judged it to be bad or wrong, arguing that in matters of faith every effort must be made to know and to answer whatever possible.
To that end, R. Gil's excerpt from RYBS (who is himself invoking Kierkegaard's Fear and Trembling) indeed demonstrates that there are instances in which the man of faith must throw up his hands and say "teiku." There are theological questions and contradictions so profound that the true man of faith must accept that some questions will not be answered.
He could have cited the example of R. Yehuda Amital as well. A cornerstone of his post-Holocaust theology was resignation without any attempt at justification. This was the basis for his critique of those - from R. Zvi Yehuda Kook to R. Yoel Teitelbaum - who rationalized the Holocaust within a larger ideological scheme (see, for example, pp. 127-133 of By Faith Alone).
Yet my critique then, as now, focused on what I believe to be a common misunderstanding of the idea of teiku. Teiku is said not when I don't know something and not even when nobody knows something. It is used when something cannot be known (note that in the RYBS quote he shifts from the real meaning of teiku to the colloquial meaning of teiku just before the end of the quote; I wouldn't put too much linguistic stock in the rhetorical jab at the end).
Resignation in the face of the unknown should never be a long-term solution. It may be prudent in some instances, but it must never become policy. It stunts all discovery and progress. At the same time, recognition that some things-like the reason for the binding of Isaac and the Holocaust-are unknowable is a desirable trait. It means knowing that man can never know the answer, or that there is no answer - not that someone of superior intellect will eventually come and answer it for me.