No, and many people make this mistake. They are completely unrelated, and you need to keep a close eye on both.*
For example, if you are an Australia/UK dual citizen who is resident in the UK, as far as the Australian immigration regulations are concerned, you can visit Australia for as long as you like.
However, you’re still subject to Australian tax regulations. If you overstay according to those, you could be liable for Australian tax on your income, no matter where it was earned – as well as being liable for UK tax. That doesn’t leave you with much. Don’t do that.
See the Tax Compliance page for more information.
* We have simplified this explanation. Many countries have tax treaties that could exempt you from double taxation. It’s always best to check with an international accountant or tax attorney if you are planning to spend considerable amount of time in another country. Our partner is Global Tax Network.