I am a UK citizen currently living in the Netherlands.
What would I have to do, or where would I have to go, in order to renew my passport?
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The UK government website has an an interactive tool for find out how to renew your British passport from outside the UK.
In practice, this may soon not be needed. Over the past few years, there were regional passport issuing offices in embassies (such as in Washington, DC, for the Americas). These are being closed and all overseas applications will be processed from a special office in the UK (in Durham).
These overseas passport applications will require a form that is different from the the UK domestic form: OS United Kingdom Passport Application.
For the most up to date information, follow the interactive tool. It may send you to the new apply for a British Passport service at HM Passport Office.
You can search online for British Consulate or Embassy near your town or region.
In UK, as a Polish national I go to Manchester Polish Consulate, pay £90, give my old passport and in 3 weeks I get a new one. But you have to go personally each time.
The same applies if you loose your Passport. But in EU you can travel with a valid photo id card now. So you can go home and do a lost passport thing there, but you DO not have to. The cost and time is much longer for lost passports in Consulates and Embassies in foreign countries and if you loose your passport again before expiry of your replacement you have to go to an embassy and explain yourself.
You can also apply for emergency passports at the consulates and embassies if you need to travel internationally within the next week. It costs more and they expire after 6 months.
I can speak from experience on this one. I jumped on the train to The Hague to see the British Embassy about an unrelated matter, thinking there was only one British Embassy - but I was wrong. They told me they only renew driving licences and/or passports there.
For anything else I was to go to the British Consulate instead, in Amsterdam.
I'd call and check, but that's my recollection.
Only a country you are a citizen of can issue you a new passport (or renew existing one).
EU is not a country, but a union of independent countries, therefore there's no such thing as EU passport. There are just national passports issues by independent countries.
You must contact your government (UK), but it's also possible, you can apply for new password in UK embassy.