Yes, the price paid by consumers will change as long as they are willing to pay the extra money for the sales tax.
In a situation where supply is perfectly elastic, suppliers are willing to supply an infinite amount of a good at a given price. They are not, however, willing to supply any of that good below that price (let’s call it Price X). If a sales tax is imposed and the price to the consumer does not go up, the suppliers are essentially selling at a price that is lower than Price X. This is because the consumer still pays Price X but now the government gets some of that money. Therefore, the suppliers will stop supplying any of the product.
Therefore, if consumers want to have any of this product, they are going to have to pay the sales tax themselves. The price that they pay is going to have to rise by exactly the amount of the sales tax. As we do not know anything about the demand curve, we do not know if all of the customers will pay this new price, but we do know the suppliers will not produce if they do not get that new price.