As a tourist in Colombia, do I need to pay the 16% hotel tax?
If not, how do I get this money back?
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Sorry for the late answer. You shouldn't pay any hotel tax in Colombia, even if they insist. Anyone traveling on a tourist visa is exempt from paying taxes on any hotel registered with the National Tourism Register (which I understand is a requirement for every hotel in the country) or a travel package. I own a tourism company in Colombia and cleared this with my accountant in our initial meeting. I've also traveled extensively in the country as a tourist before moving here and have never paid hotel tax. If IVA shows up on the bill at checkout, ask them to waive it and they usually do it without hesitating.
I've just been able to find references in Spanish but it seems that you don't have to pay that tax. Here is a website explaining general taxes in Colombia. The tax you're referring to would be the IVA (VAT in english).
If you google something like "IVA hoteles Colombia" (VAT hotels Colombia), you can find numerous websites stating that foreigners do not need to pay this tax if they have purchased some kind of holiday package like this one or this one (both of them in spanish). In case you don't have a package like this it seems that you should pay the tax anyway.