Surprisingly, Google shows "he took her out for dinner" is used just as much as, if not more, than "he took her out to dinner".
I think "out to dinner", as in "he took her out to dinner" sounds better. I would say it sounds smarter, though I can't say exactly why. It just does.
for dinner - the purpose of eating dinner - mmm...
The phrase "out for dinner" works better as part of another idea or a different sentence.
Student or Learner