Applying for scholarships for PhD before I know where (and which country) I'll be accepted
since the US has a reputation here in Europe to have very high tuition fees. Do I understand correctly from this statement that Harvard Economics PhD students don't have to pay any kind of fee to the university for the full 5 years, or am I missing something?


I am a graduate student of economics in continental Europe, so I do not have a first-hand experience about the US or the UK. However, my general impression about the field is that those very high tuition fees are more relevant in undergraduate degrees and professional non-PhD degrees (such as MBA, terminal applied MSc, etc.) and not much about the PhD path.

I do not know for sure about the Harvard PhD funding policy, but your interpretation is certainly plausible. The general expectation among the PhD students is to receive some funding, and actually having to pay tuition to the school (in addition to not getting any living subsidies) seems unusual (and inadvisable).

The passage below from the econphd.net guide supports my impression that actually paying tuition  should be more of the exception than the norm for PhD students.


  Economics departments give considerably lower fellowships than business schools or science departments. Worse yet, some people will be admitted without fellowships and a few end up paying tuition. But it is commonly understood (although the department may not be willing to explicitly guarantee it) that all students who stay on beyond the first one or two years are funded in some manner. Often teaching assistantships are part of it, but many programs also reward academic success with scholarship money.
  
  Generally, the best and most competitive programs offer the most generous funding – MIT, Princeton, Harvard, Stanford, and Yale admits can expect to get a package of, perhaps, between $ 16,000 and $ 20,000 per year (with tuition waiver).


That said, I believe there are also externally funded PhD students (who receive funding from governments, other research institutes, grants, etc.) Having your own funding may help in the admission, but I do not know the process or how normal it is.