Typically, in the United States (I am not sure which country your friend's passport is from), there is not a problem so long as the chip is working. I'm confident that is the same in Europe, as I've traveled throughout with what you're describing. [I hold a USA passport with a work visa in the UAE]
When I took the same, worn passport traveling to other countries, it caused a lot of trouble, even though the numbers, the photograph and the bar code were both legible, and the chip works.
- In Saudi Arabia, I almost had my transit visa taken away, because they didn't 'like' my passport.
- In Nepal, I waited hours in a wooden chair at the airport in Kathmandu. It was not explained to me the issue specifically, but it was about my passport, so I am assuming it was the presentation of it, since I don't really have anything controversial with it.
- In Israel I was subject to extreme scrutiny and questioning (before departing, oddly), but I will caveat that this could have been because I have a lot of stamps from countries that they politically do not tend to favor. Nonetheless, the passport was again the issue.
- In the United Arab Emirates, I was detained in the airport overnight and missed a day of work, because of the condition of the passport - being told after my PRO talked to them, that I "need to get a new passport."
My very strong recommendation is to have your friend get a replacement passport just to avoid the unnecessary hassle if someone doesn't like how worn or 'suspicious' it looks. The minute I got back to Abu Dhabi I went to the US Embassy, got a replacement, and haven't had a problem since.
So she might not have a problem, but it's probably not worth the risk.