There are, in fact, many different theories purporting to "prove" a Phoenician presence in the Americas. Are we talking about Berry Fell claiming to have found "Punic" writing on a stone in Massachusetts? Possible Carthaginian coins in Alabama? Amphorae in a Brazil bay? Or maybe a rock in Brazil with purported Phoenician writing? Without additional specifics as to what you are referring to, I'm not sure I can really give a thorough debunking.

And any discussion would be a debunking, because there is not actually any real proof of Trans-Atlantic Phoenician travel. Objects like the Bourne Stone (and various other inscribed rocks) have been shown to be either fakes (Bat Creek Stone, Grave Creek Stone) or misinterpreted Indigenous petroglyphs (Bourne Stone). Of the multitude of ancient coins found in the Americas, many are not actually ancient but modern products, and they never seem to manage to get found in proper archaeological context, as detailed in Epstein (1980) "Pre-Columbian Old World Coins in America: An Examination of the Evidence."

Even one of the biggest proponents of Carthaginian coins, geologist Mark Menamin, had to eventually conclude the "Farley Coins" which he and other diffusionists put so much stock into were either fakes or modern reproductions of ancient coins. In his book, Phoenicians, Fakes, and Barry Fell, he concludes that cache of coins "may no longer be used to support the early crossing hypothesis.

The amphorae found in Brazil which may have been Phoenician (though most often claimed to be Roman)? This was claimed by a professional treasure hunter who, once he was denied funding, claimed a cover-up, conveniently not mentioning that he already in trouble for clandestine sales of other objects he had previously treasure hunted. Also, a local businessman stated he put the jars there to artificially age them, and that style of jar was used, and locally manufactured, well into the Colonial era.

As for the Paraiba Stone which was claimed to have a long Phoenician inscription about a Phoenician king? No one ever actually saw the stone, which mysteriously disappeared after being transcribed, and the transcription itself has been shown to be easily forgeable.

So could the Phoenicians cross the Atlantic be true? Maybe, anything is possible and there was a modern replica Carthaginian ship that a team sailed around Africa in order to re-create an ancient voyage which may or may not have happened (per /u/mythoplokos). The problem with modern fabrications of ancient ships being used to "prove" ancient peoples sailed to the Americas always has the intrinsic problem of explaining why ancient mariners would want to sail off into a vast empty ocean with no idea what was on the other side, or even if there was anything out there. We modern people benefit from a well-mapped world that we can literally see pictures of from space. We know the boundaries of our planet and the shape of its geography in a way unfathomable to an ancient person.

We can envision an Atlantic crossing because we know the Atlantic is crossable and the Americas await us on the other side. Meanwhile an ancient person, even one with a mathematical understanding of the size of the globe, has no such guarantee. Indeed, any ancient person with an understanding of the size of the planet would be more intimidated by sailing westward, not less. The proposal of a trans-Atlantic crossing becomes less appealing and rational if you think of it in terms of sailing into a vast uncharted ocean for months with the baseless hope of finding something along the way to prevent a horrible death of dehydration and starvation before reaching China.

So as for evidence to back up the Phoenician claims? None.