If the whole sentence referred to present time, you'd use 'doesn't' or 'won't':
Facebook is facing intense pressure to ensure it doesn't happen again.
or
Facebook is facing intense pressure to ensure it won't happen again.
The pressure is happening now, and the 'doesn't' and 'won't' both refer to the future.
In your sentence, we use the past tense 'Facebook faced' because this is something that happened three years ago. So the rest of the sentence has to follow on from this, also using a tense which relates to past time: in this case, 'the future in the past'. So equivalent tenses here are 'didn't' and 'wouldn't'.
Facebook faced intense pressure to ensure it didn't happen again.
or
Facebook faced intense pressure to ensure it wouldn't happen again.
The pressure happened in 2016, and the 'didn't' and 'wouldn't' both refer to the 'future in the past'.