However, "Eat the dinner" sounds unnatural. Are you sure she didn't say "Eat your dinner"?
We use "to eat" in many different ways. Here are just a few random examples:
I was eating lunch when my house fell down.
What do you want to eat tonight?
Where are we eating on Saturday evening?
I don't eat peas.
Eat up before it gets cold.
The guilt is eating me alive. (Metaphorical use)
I haven't eaten for days.
He died from eating bad mussels.