The Supergiants had earlier lost Kevin Pietersen, Faf du Plessis and Australia all-rounder Mitchell Marsh to injuries.
Smith's pull out was confirmed by Cricket Australia Sports Science and Sports Medicine Manager Alex Kountouris, who said the right-handed batsman had been struggling with pain in his right wrist for the last week.
"We have been working with his IPL franchise Pune to monitor this and unfortunately it has not resolved," Kountouris said in a statement from Australia.
The CA medical official stressed that the pull out decision was taken as a precautionary measure and the injury did not appear to be serious.
""From the information we have so far, this doesn't appear to be a serious injury but we are keen to give him the time to recover and rehabilitate before heading to the West Indies at the end of May," Kountouris said.
Australia begin the tri-series in Guyana on June 6 against South Africa before facing West Indies four days later at the same venue.
Smith's pull out comes a day after Australian all-rounder Mitchell Marsh was ruled out of the rest of the IPL with a side strain.
The Mahendra Singh Dhoni-led Supergiants are currently languishing in the bottom third of the points table with just two wins.
With injuries plaguing their foreign recruits, the Pune outfit is now desperate to get Australia batsman Usman Khawaja into the country. Khawaja was roped in up after the loss of Pietersen and du Plessis.
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