According to Bloomberg:
In one of the first efforts to quantify the impact of states accruing more tax revenue from Web purchases, researchers at Ohio State University published a paper this month that found sales dropped for Amazon when the online charge was introduced. In states that have the tax, households reduced their spending on Amazon by about 10 percent compared to those in states that don’t have the levy. For online purchases of more than $300, sales fell by 24 percent, according to the report titled “The Amazon Tax.”
You can find the paper here (PDF), but I'm not sure that there is really any need to read it.
That paper proves a basic theory of economics that we all learned in Econ 101, namely that demand goes down as prices increase. It's called the demand curve, and that more than anything explains why Amazon fought the various state's efforts to collect a sales tax on online sales. This money was never going to come out of Amazon's pocket, so it wasn't going to hurt them directly, but Amazon knew that the taxes would discourage sales.
Update: It looks like I wrote too soon. This isn't an example of consumers spending less; they are just more choosy about where they are shopping. Here's a telling quote:
When we look at the sales of Amazon Marketplace merchants, who are generally not subject to the Amazon Tax, the large sales (? $300) of these retailers increase by 60.5% after the tax goes into effect. We conclude that to a small degree, the tax legislation achieved its objective of restoring retail activity to local communities, though most of the gains in “leveling the playing field” are garnered by the online operations of retailers.
Overall, our study shows that Amazon experiences a decline in sales following the implementation of an Amazon Tax. Households substitute Amazon with other retailers: either online retailers who are exempt from collecting sales tax, or in-state retailers (online and brick-and-mortar).
I will read further, but one effect of the Amazon tax is to shift sales from Amazon to their Marketplace (where Amazon charges a commission). In other words, Amazon is still profiting from tax-free online sales.
And this still proves a lesson from Econ 101: consumers shop around for the cheapest price.