There are no EU-wide rules that say how EU nationals who live, work or spend time outside their home countries are to be taxed on their income - coming from wages, pensions, benefits, property, successions and donations, or any other sources.
There are only national laws and bilateral tax treaties between countries - and these don't cover all eventualities and vary considerably.
However, there are some basic principles that apply in most cases to people who spend time in EU countries outside their home country:
Other taxes
For information about taxes on unearned income in the country where you are tax resident (e.g. property taxes, local taxes, gift and inheritance taxes, etc.) consult the local tax office.