Is there an advantage in including quotes from teaching evaluations in a teaching statement?
I would consider student comments in teaching statements as "standard optional": they are not required, and probably the majority of teaching statements do not include them, but a substantial minority do and it is certainly quite reasonable to do so.  Here are some things to keep in mind:


Your teaching statement is your chance to convey your philosophy of teaching and, in so doing, show yourself to be a clear thinker, competent writer and to possess fluency in English (assuming that you are applying to a job where English is spoken).  Including too many quotes by other people in a short statement works against this.  I am a wordy person, but for me I think a teaching statement should be at least two pages long.  If you have a 1.5 page teaching statement and a third of it is quotes, then I worry that you're using the quotes to evade the short essay you've been asked to write.  Succeeding in an academic job is a lot harder than succeeding in writing a two-page essay, so that doesn't inspire my confidence.
The positive side of the last comment is that it would be much better not to just include quotes in a pile at the end but rather to use them to illustrate points about your own teaching.  Drawing an explicit moral from each quote would not be overdoing it.
Including quotes in your own teaching statement is a bit like including quotes on the back of a book saying how good it is.  These quotes can be part of a successful sales pitch, but let's be real: they are cherry-picked.  (If you're not cherry-picking the very best quotes, you're losing ground to everyone else who is. In particular, think ten times before including a quote that has the slightest hint of negativity or even measured praise about your teaching.  Saying something negative about yourself in application materials is a black belt level sacrifice throw.)  The idea that you fundamentally gain legitimacy from a teaching statement by including the students' own words is a dubious one to the jaded eye.  Most of the eyes reading your application will be highly jaded. 
Consider feeding the quotes (especially if you have more than one or two) to a faculty member who can put them in a teaching letter.  It is harder to write good teaching letters than good teaching statements.  Along with cold, hard statistics about your teaching (but be careful about introducing negativity, as above), including quotes from students is a good way to personalize the letter.  (Saying that Ms. X is clear, attentive and well-liked by students only fills up so much space, and almost everyone else is saying it too.)