Jon Tan, in a recent article at 24 Ways, talks about the science behind the positive effects of good typography. In his article, he references a study by Dr Kevin Larson of Microsoft and Dr Rosalind Picard of MIT that explored the effects of good typography on mood:
Two versions of the New Yorker ePeriodical were created. One was typeset well and the other poorly … [the doctors] engaged twenty volunteers — half male, half female — and showed the good version to half of the participants. The other half saw the poor version. The doctors found that, “there are important differences between good and poor typography that appear to have little effect on common performance measures such as reading speed and comprehension.” In short, good typography didn’t help people read faster or comprehend better … [the doctors] also found that “the participants who received the good typography performed better on relative subjective duration and on certain cognitive tasks”, and that “good typography induces a good mood.”
In summary, between the volunteers who read the bad typographyical layout and those that read the good typographographical layout, there were no differences in reading speed or comprehension. However, those volunteers who read the good typographical layout perceived it as an easier task and were induced to a good mood.
It can thus be inferred that setting type without regard to contrast, font size, kearning, leading, and so forth, is still functional. It still works. People can still read bad typographic layouts without much difference in "reading speed and comprehension".
Hence, taking the time to incorporate principles of good typography in your designs won't necessarily improve readers' comprehension. It will, however, leave people feeling good (though they may not know why). And as web designers, that is a goal worth striving for! We are not creating software. We are not making things beautiful. We are designing experiences. Expereinces that leave people feeling good.
And if inducing a good mood isn't reason enough to invest in good typography, the study also proved that good typography made people "more capable of completeing creative tasks faster".
So if your app helps people accomplish tasks, especially creative ones, you had better consider the positive effects of great typography!