R' Chaim B. of DivreiChaim was recently pointing to several dangers of rationalism:
Taking it too far - Some people take rationalist methodology too far and end up denying ikkarei emunah.
Too much independence - The rationalist push towards people making their own analyses and drawing their own conclusions means that sometimes, even those who are not qualified to do so make their own (faulty) conclusions.
Weakening respect for Torah sages - By showing that they could have erred scientifically, the rationalist approach risks undermining our respect for Torah sages.
I completely agree with R' Chaim's concerns. They are absolutely valid and I have pointed them out in my books. In fact, it is precisely for these reasons that I do not try to make non-rationalists into rationalists. (I see my mission as showing closet rationalists that there is a place for them in Judaism.)
However, for the sake of balance, I think that it is important to show the other side of the picture - the dangers of anti-rationalism.
Taking it too far - Some people take mystical methodology too far and end up denying ikkarei emunah, e.g. by effectively attributing independent power to different mystical forces, or making God into discrete parts. Behind all the flowery terminology, it's polytheism. Alternately/ additionally, some people end up denying obvious scientific facts, e.g. the shape of the world, the number of teeth that non-Jews have, spontaneous generation, which can drive people away from Judaism.
Too little independence - The non-rationalist push towards people placing their trust in others means that sometimes, they end up placing their trust in those who make faulty conclusions and there is no way of their being corrected.
Weakening respect for Torah sages - By claiming that they were scientifically omniscient, the non-rationalist approach risks fundamentally undermining people's respect for them when they find out that this simply isn't true; they have been led to believe that being scientifically infallible is an integral part of their greatness, and if they are proved scientifically fallible, it means that they were not great.