There are some cases where repeating the article can prevent unnecessary confusion—or make something explicitly clear.
When she went to the hospital, she picked up three things: a contagious disease, prescription, and bill.
In this sentence (although common sense would tell us not to interpret it in such a way) the word contagious could be taken to apply to all three items—indicating that not only the disease but also the prescription and bill were contagious.
In order to explicitly avoid applying contagious to all three items, you would remove the ellipsis used in the construction by repeating the a in front of each item:
When she went to the hospital, she picked up three things: a contagious disease, a prescription, and a bill.
Because a is repeated for each item, and contagious is only used with one item, it doesn't require any parsing to understand that neither the prescription nor the bill was contagious.
On the other hand, if you do want to relay the fact that all items were contagious (against common sense), you would make the same explicit adjustment:
When she went to the hospital, she picked up three things: a contagious disease, a contagious prescription, and a contagious bill.
By using a and the adjective in front of all three items, you force a particular interpretation.
For similar reasons, it makes parsing a sentence that has multiple items, each with its own adjective, a little clearer.
Consider this sentence:
I ate a delicious red apple, rotten green apple, and mealy yellow apple.
We assume that only the first apple is delicious.
But now read it when that's made explicit:
I ate a delicious red apple, a rotten green apple, and a mealy yellow apple.
By using a in front of each of the items, it doesn't leave you having to parse, and reject, the applicability of the first item's adjective to the remaining items.
For instance, we know (or assume) that the original version of the sentence isn't actually saying this:
I ate a delicious red apple, a delicious rotten green apple, and a delicious mealy yellow apple.
We know this because common sense tells us that wouldn't be the case. But using the article prevents us from having to apply common sense in the first place.
And, as with the example of the hospital visit, if somebody actually did want to express such a sentiment contrary to common sense, they would do so in this explicit fashion.
But if adjectives aren't used, and it's a simple construction, omitting all but the first article can make the sentence flow better:
He ate an apple, pear, and banana.
This is shorter and more succinct than the following:
He ate an apple, a pear, and a banana.
However, based on personal preference and style, you might think that specifying an with the first noun and a with the remaining nouns is actually the better sounding sentence.
Sometimes repeating a helps prevent misunderstanding—especially when adjectives are used. At other times, there is very little likelihood of misunderstanding, and omitting a from all but the first item makes the sentence flow better.
It's up to context, style, and preference if you want to (or should) repeat articles.