It certainly does! In Canada, for example, you have to be a resident _somewhere_. So you cannot just move around from country to country on tourist visas (which obviously do not permit you to declare residency) and claim to the Canadian Tax authorities that you are a non-resident and avoid paying all taxes.
Indeed. In Europe Monaco is a popular place for very rich people to (pretend to) live due to the lack of income tax. You are restricted in how long you can spend in other countries before being deemed to be domiciled there for tax purposes. In Ireland and I think the UK also, it's 183 days a year. And they do occasionally check, if you're rich enough.