No, "Biology is more interesting than literature is more interesting" is not grammatical.* Repeating the word "more" is not right.
The basic pattern
This sentence illustrates the basic pattern, though it's not how we usually say it:
Light travels faster than sound travels.
There is only one comparison: "faster than". There are two things being compared: how fast light travels and how fast sound travels. We nearly always elide the second verb except in the rare situation where the verbs are different. So, this is the normal way to form the sentence:
Light travels faster than sound.
Here are some examples with "more":
Light travels more quickly than sound.
Iron is more useful than gold.
Gold is worth more than iron.
The entire phrase starting with "more", including the subordinate clause starting with "than", functions as an adverb (as on "travels") or complement.
The correct answer
So, a correct answer would be:
Biology is more interesting than literature.
The complement of "biology" is "more interesting than literature", not "more interesting".
You could also say either of these, which include one or both of the elided words, but these are unusual; normally we would say them only to make some unusual emphasis:
Biology is more interesting than literature is.
Biology is more interesting than literature is interesting.
A few more examples
Here's an example of the full sentence pattern, with two different verbs and two objects:
Water flows downhill faster than salmon swim upstream.
And here are a couple examples with two linking verbs:
The clown looks happier than he feels.
The doorman is more useful than he looks.
Most common, though, is the simplest pattern, illustrated by these well known sentences:
That dog's bark is worse than his bite.
Blood is thicker than water. (A traditional proverb.)
A good name is worth more than silver or gold. (A Biblical proverb.)
Imagination is more important than knowledge. —Albert Einstein
The pen is mightier than the sword. —Edward Bulwer-Lytton
* Technically, it could be grammatical if set up in a very unusual context, but that's too strange to bother with when mastering the basics.