Does Daylight Saving Time affect the flight time?
It will affect scheduled flight times the same way it will affect your favorite TV show's air time.

i.e. it won't*

It will affect apparent flight duration if the departure and arrival airports switch to DST on different days of the year, but actual time in the air will remain the same.

* The caveat for that is if you happen to be in an area that A) does not switch to DST and B) your favorite TV show happens to be broadcast from an area that does switch to DST. (This situation happened every year in Indiana for many years until IN joined the 19th** century and switched to DST in the summer.)

** Yes I know that DST was a 20th century invention, it's just that Indiana is a bit behind the times...