What is a sentence beginning with "yes" called?
I'd suggest Affirmative, if we ignore the lyric, "Yes, we have no bananas."
I thought maybe an interjection?
No: interjections are words like "ah, gosh, hey, oh, ooh, ouch, whoa, wow" and so on. But not "yes".
Yes and no answers.
"Yes" and "no" serve as markers of positive and negative polarity in answers to questions. They can stand alone, or combine with a clause to express the answer more explicitly.
"Is this car yours"?
"Yes (it is). / "No (it isn't")
In examples like these, "yes" and "no" are ADVERBS.
Does that help?