There's simply no evidence to support that claim and indeed evidence to the contrary. This is just misinformation spread by the zinc lobby, and I'm sorry that I have to tell you that, yes, the zinc industry is paying money to gaslight you into keeping their useless product around too. That is, this is zinc industry FUD.

• In 2022, the US Mint produced 6.36 billion pennies and 4.52 billion in 2023 (the latter is a huge low) for a total of 10.88 billion pennies in two years. If we multiply that by its current cost to print of about 3¢, we come upon a figure of about 326.4 million dollars spint minting just the penny over two years, or nearly 50¢ per citizen per year.
• In Canada, prices are rounded both down and up and have been for 12 years, and there's been no evidence of "gouging".
• As noted, a multi-state 2006 survey showed that not only were prices not higher for consumers per transaction due to rounding but that in fact, they gained an extremely marginal 1/40 of a cent on average per transaction.

With these three facts, we're already at a point where you're gaining nothing per transaction by keeping the penny and you're in a relatively steep penny debt each year by nature of them being minted. However, let's continue.

If you routinely do cash transactions, try to consider how much of a cumulative hassle and overall cost pennies make things outside of just minting them.

• If you're getting pennies as change (which will happen as long as you're not paying exactly with pennies and it isn't a multiple of 5 cents), then you have a few options: a) "Keep the pennies" which loses you the pennies, saves some time, but still incurs that slight delay where the cashier needs to mentally process this. Or b) Waiting for the cashier to get the pennies out of the tray. If you value your time at a pretty measly $10/hr, even just two seconds of additional waiting for and handling the change loses you about half a cent. You also have to account for the fact that the penny makes it less likely that you'll want to go through the trouble of having or counting out exact change, and so with the penny there's more risk that this relatively lengthy transaction occurs where the cashier counts out your change, hands it back to you, and you put it away.
• Now consider the hassle if you try to have exact change. You now need to carry around this other type of coin, find and count out enough of them to present to the cashier, and then wait for them to sort out the pennies to place into the appropriate tray. That's almost assuredly multiple seconds lost.
• Rounding to the nearest five cents massively increases the likelihood that you land on a cleaner denomination. Transactions which are $X.01, $X.02, $X.98, and $X.99 now get rounded to the nearest dollar, meaning no change at all is involved. Because previously those values were limited to $X.00 and $X+1.00, this effectively triples your chances of landing on a clean dollar value. For transactions which require quarters as the smallest denomination, this takes it from a 3/100 chance for $X.25, $X.50, and $X.75 and quintuples it to 15/100 by adding in $X.23, $X.24, $X.26, $X.27, $X.48, $X.49, $X.51, $X.52, $X.73, $X.74, $X.76, and $X.77.
• These businesses need to go through the process of handling the pennies too, and if the volume of transactions is high enough (say in a grocery store), that is a marginal factor into what you're paying.
• The negative climate externalities of producing the penny aren't factored into the cost to produce the penny. Their carbon footprint for the most part will nonetheless simply be passed onto the people (us) who have to deal with climate change. At 10.88 billion pennies, this is over 27,000 metric tons of zinc over the last two years, and while there are plenty of issues that dwarf that in absolute terms, relative to whatever "benefit" it confers (it's actually a detriment), it's pretty bad. There's of course the fact that there are billions of pennies right now sitting out there as litter to consider as well.
• Modern vending machines (which almost universally also now take cards as payment, but are probably still one of the more likely places you'd want to use cash) have literally no option for a penny because of how grossly impractical they are.
• Unless you routinely count out exact change, you're likely to have more pennies than you'll actually be spending, and at that point, you just have pennies sitting around depreciating in value and doing nothing as opposed to at least offering you some utility. (There are even multiple Reddit threads on frugality about how you can get rid of bulk pennies at banks or by using them at the self-checkout.)

Pennies are pretty unambiguously a net detriment to everyone except the zinc industry. The one being gouged right now is you by being forced to subsidize the zinc industry's less-than-useless product.