I hope this will explain everything. In spoken language you can use her instead of she, but in official writing this word must match the verb of the sentence. So in the sentence: I have more than her, her is informal way of what one should officially write as I have more than she. (which is abbreviated form of I have more than she has.).
As PatHargan here writes:<pre>
The distinction is just the same as in the Irish examples.</pre>
Then it works both ways.
So: Tá níos mó agam ná (mar atá) aici. is literally: I have more than she (has). and Tá níos mó agam ná í. is like saying I have more than her. which has double meaning:
1) I have more than her (she’s the object of having, a possession)
2) I have more than her (unofficial way of saying I have more than she has).
In English, 'I have more than she' and 'I have more than her' are both grammatically correct (if not necessarily idiomatic), but have different meanings: in the first sentence, 'she' is the subject of an implicit 'has' - i.e. she has less than I have - whereas, in the second, 'her' is the object of 'I have' - i.e. I have not only her, but others besides. The distinction is just the same as in the Irish examples.
I put "I have more than she", which was marked wrong, but Duo has given for one "correct" answer to this, "I have more than she's". I can only think that, in this case, "she's" means "she has." (I would never say it this way as a Canadian English speaker, not in that spot in a sentence, but maybe some people do - in any case, I can't see it meaning, "she is".) In other words, Duo seems to be accepting "I have more than she has" as a possible acceptable answer (albeit in what it, to me, a very odd way of saying it) which is, in my mind, the same as "I have more than she."
So, something about this needs to be fixed, either in the Irish or the English.
 I got this one a second time, and I typed, "I have more than she does", and it was accepted.
I think this "I have more than her" needs more to the sentence to understand. They are saying something like "I have more (friends-more than just her) than her." Not I have more of (something) than her. Which is odd for us Americans because we never use "I have more than her" in the first example here but always in the last one.
A correct answer, although not the most common is, 'I have more than she'.Her is also correct, it indicates that the English language is always evolving. Her is, of course, the object form. It's common use in comparatives, is an example of the fact that in the English language if a mistake is made often enough it can become the norm. That does not mean that older, more correct usages become incorrect. As long as they can be understood, they must be accepted.