No — citizenship and tax residency are separate concepts. Receiving a St Lucia passport does not, by itself, make you a St Lucian tax resident. St Lucia (and most Caribbean CBI states) operate territorial tax systems that only apply to physical residents and locally-sourced income. A CBI holder who lives, works, and earns abroad is generally outside St Lucia's tax net.
Your Australian tax position is governed by Australian residency rules — primarily the resides test and the 183-day test. Holding a second passport doesn't affect those tests directly. What changes things is where you actually live: if you eventually relocate and break Australian tax residence, then the rules of your new home country apply. We strongly recommend coordinating with both an Australian tax adviser and a tax professional in any country you plan to spend material time. Citizenship is the easy part; tax residency is the part that needs careful planning.
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