If I open a bank account in a tax haven country, I don't have to pay any tax on interest.
Most countries tax your worldwide income meaning that regardless of where your income originates, it's still subject to domestic taxes. If you fail to disclose your foreign holdings, you may have to pay a penalty (ex: FBAR in the US). There are some exceptions - for example, under a territorial tax, your foreign passive income is exempt, and under remittance-based tax, you don't pay the tax if you don't (or until you do) remit foreign income into your country.
If I open a company in a tax haven country, I don't have to pay any corporate tax.
In almost any country, your company is a tax resident if it has its mind, management and control there. If you are the sole shareholder, and you live and conduct business in your home country, your company is also a tax resident in that country. It usually doesn't matter where the business is registered, and those countries that only consider the place of registration (ex: Bulgaria) also tend to have CFC and other anti-avoidance rules.
If I live in a territorial-tax country and work for foreign clients, I don't have to pay any income tax.
Territorial-tax countries like Panama and Georgia still tax your local-sourced income. It doesn't matter where the clients are. If you live and work in a country, then your income is sourced locally and is thus subject to the local income tax. For individuals, the territorial tax exemption mostly applies to passive income from abroad - it does not apply to active earned income. Failure to disclose this income would constitute tax fraud.
Once I am a tax resident in a tax haven country, I can live and work abroad tax-free.
Some countries like Cyprus offer tax residency if you spend a few weeks or months there. Unfortunately, being a tax resident in one of these countries, and even having a tax ID or tax certificate, does not prevent you from also becoming a tax resident in another country where you spend the rest of the year. Unless there is a double-tax treaty between the two countries, you will be liable for taxes in both.
If I stay less than 183 days in a country, I don't have to pay any tax.
The 183-day rule is one of many in determining one's tax residency status. Most countries consider not only the time you spend there, but also the ties you have with that country. These include residential ties such as your primary home, spose/common-law partner and/or dependents, as well as your center of vital interests and habitual abode. For instance, if your primary home is in Australia but you spend most of the year traveling abroad, you would still be deemed an Australian tax resident, even if you spend less than 183 days there. As you'll see below, even if you're not a tax resident in a country, you may still owe taxes there.
If I'm not a tax resident in a country, I can live and work there without paying any tax.
Most countries levy non-resident withholding tax on income that is sourced locally. As such, you do not have to be a tax resident in a foreign country to owe taxes there. For instance, if you work remotely in a foreign country as a non-resident, your work will be local sourced and thus subject to the non-resident WTH tax. Often though, the DTT with your home country will overwrite the local tax code, and you will only pay taxes to one of the two countries, the one in which you have your permanent home or other ties. However, if there isn't a DTT between them, you may owe taxes to both.
If I live 4 months in 3 different countries, I don't have to pay taxes anywhere.
In theory, it's possible to not be a tax resident of any country. In practice, you are most certainly a tax resident somewhere depending on the laws of the countries you visit and the country you are domiciled in. Typically, your tax residency is in the country you spend the most time in and/or have the most ties with (family, property, bank accounts, etc.). Otherwise, you might default to being a tax resident of your country of citizenship. Being a tax resident of no country also complicates banking and investing, as you're still liable for WTH tax.
If I live in a territorial-tax country, I can earn passive income from abroad tax-free.
When you earn income that is sourced in a foreign country, that country will usually deduct a non-resident withholding tax before the payment is even received. For example, if a US company pays you, a foreign person, a dividend, it will withhold 30% from the distribution (thus diminishing your dividend yield). Although this tax is non-refundable, it's usually reduced by a tax treaty with your home country (ex: from 30% to 15%). If your country then taxes those dividends again as worldwide income, you can offset double taxation by claiming foreign tax credits on your tax return.
Once I am no longer a tax resident of my home country, I don't have to pay taxes there.
Your home country will still levy non-resident withholding taxes on any income sourced locally. For example, as a tax non-resident, if you keep a rental property when you leave Canada, your rental income will be subject to the 25% withholding tax. This tax also applies to any other local-sourced income like interest or dividends (although the rate on those may be reduced with a tax treaty). You may also need to file a non-resident income tax return to report this income.
If I invest in stocks through my offshore company, I can pay a lower withholding tax on dividends.
Most countries have anti-treaty shopping provisions, in particular, limitation on benefits clauses. If you want to benefit from a reduced WTH tax rate in a foreign country, you will need to establish economic substance there.