"Dine" does suggest something rather more formal/substantial than "eat" (you can only use it for a main meal, for example, not a snack), but I don't see why you cannot say "dine in fast food restaurants ".
From the example sentence the OP gave, it is clear that the question is not about the phrasal verb "dine in", but about the use where "in" is a preposition.
Interestingly, though, the phrasal "dine in" doesn't always mean "staying home to eat", but can also be applied to restaurants that offer the choice between eating on the premises and a carry-out option.
In some chip shops they have a few tables and chairs available, and when you place your order at the counter, they might ask you "Is it to eat in or take away?".
Personally, I wouldn't use "dine" for this. The term seems to require a certain modicum of salubriousness.
I would "dine" in a slow-food restaurant, but "eat" in a fast-food restaurant. Indeed, I'm not even sure that a fast-food place deserves to be called a "restaurant"!