Long long ago in the dense forest lived a lion.
Long long ago in the dense forest there lived a lion.
Are the both correct?
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Once upon a time there lived* a lion in a forest.
English clauses which are not imperatives** must have a subject. Sometimes we need to use a ‘dummy’ or ‘empty’ or ‘artificial’ subject when there is no subject attached to the verb, and where the real subject is somewhere else in the clause. There is one of the two dummy subjects used in English.
Regarding the first sentence, it’s also correct. This grammatical construction is called subject–verb inversion.
Beside the bed stood a lamp.
Down the street lived the man and his wife without anyone suspecting that they were really spies for a foreign power.
In the vase are some flowers.
On a side note, subject-verb inversion is usually impossible if the subject is a weak (non-stressed) definite pronoun. You can’t write
Long long ago in the dense forest lived it.
*It’s a common literary technique used to begin some children's stories written in a traditional style.
There once lived a poor widow who had a beautiful daughter.
At the edge of a great forest there once lived a king with his beautiful daughters.
There once lived a woman who was so fat, she couldn't fit in a taxi.
**An imperative sentence/clause gives a command/request to do something:
 Long long ago in the dense forest lived a lion.
 Long long ago in the dense forest there lived a lion.
They are both OK.
In  the subject is "a lion", which has been postposed to a position after the verb, cf. a lion lived. A postposed element occurs in a position that tends to receive greater phonological prominence; the emphasis here is on the lion, not on when or where it lived.
In  "there lived a lion" is a presentational clause with "there" as subject.