No.  In fact, it's been a huge PITA for us infrastructure-builders.  

Online bill-paying doesn't really save car trips.  Few people actually drive to the power company to pay the bill now.  They mail it.  And the mailman comes every day, even if you don't mail anything.  

Online shopping ends up using more oil, not less.  Instead of trucks delivering to malls or Main Streets, and people driving their cars there to do their weekly shopping, you have huge trucks making daily door-to-door deliveries on residential streets.  This creates additional wear and tear on the roads, and more congestion because the local roads were never meant to handle that kind of traffic.  

The only way online shopping saves energy is if you buy a downloadable product.  A song, software, etc.

Interesting. Your point is accurate and does show that my initial reaction wasn't as well thought out as it could be. However, I expect this is fairly complex and that there are both savings and additional waste associated with on line transactions.

If I buy an item on line, yes it is shipped to me, not the mall. This uses more energy. But then again, I don't go to the mall. Over time this could reduce the need for big stores and infrastructure, which also use energy. But if I have three items shipped to me separately, this could involve three trucks coming to my house, but would have only been one trip to the mall. However, maybe Federal Express was going to deliver to my neghbor anyways.....

I agree that the only way that on line shopping is a pure energy reducer is downloadable products. However, it does seem clear that in other catagories the impact is complex and difficult to measure.

Does anyone know if this has been studied?

It has been studied, but I don't know if the information is on the Web.  The federal and state DOTs have studied it, as part of their planning processes.  
I'd imagine the present level of product availability probably requires a lot more intermediate storage and transfer facilities.
Just-in-time delivery, "rolling warehouses," etc., are distribution systems designed around cheap energy.  (Wal-Mart has perfected this system.) It works because it's cheaper to send a truck out every day than to rent more storage space on-site.  As fuel prices rise, I expect that to change.  
I need to add that it has very often been the case that I order something on-line and it is delivered at time where there is nobody at home. The resulting trip to UPS or FedEx (usually at rush hour time) kills off all possible energy savings and if the item is not that expensive, sometimes kills the price differences too.
Please explain how a handful of delivery vehicles moving through a neighborhood each day for deliveries is more traffic than every single homeowner having 2-3 cars all of which are in circulation on those same residential roads each day.

By your argument, it would be better for everyone to drive to the post office to pick up their mail rather than having the post office deliver it. I don't buy that but if you have data that somehow validates this position, I'd love to hear it.

Please explain how a handful of delivery vehicles moving through a neighborhood each day for deliveries is more traffic than every single homeowner having 2-3 cars all of which are in circulation on those same residential roads each day.

The trucks are not a replacement for ordinary traffic.  They are in addition to ordinary traffic.  

Moreover, residential streets were not designed for trucks.  They are designed for cars.  The turns are often so tight trucks can't make them without crossing over into the wrong lane or climbing the curb.  The asphalt and subbase are thinner, and don't bear up under the load as well.  A parked truck on a residential street can bring traffic to standstill; a car wouldn't be parked there, or would be easy to drive around.    

By your argument, it would be better for everyone to drive to the post office to pick up their mail rather than having the post office deliver it.

I think we may be going back to that.  That is how it was for me, growing up in a small town.  UPS wouldn't deliver; too small a market.  The USPS would deliver, but only to the post office.  People had PO boxes and picked up their mail in town.  It didn't take as much gas you might think.  Most people didn't check their mail every day.  And everyone knew everyone, so you could have a neighbor pick your mail up if they were going into town.  Many people had their mail delivered to their office PO Box.  A secretary would go down to the post office daily, and pick up everyone's mail.

I'm not convinced and would like to see some data.  Googling "energy efficiency online shopping" turned up lots of links to efficient appliances but only this item on the topic at hand:

For example, for each book sold, the online retailer Amazon.com uses just one-sixteenth the energy to operate its buildings that a traditional bookseller uses. Internet shopping also uses less energy to get a package to your house. Shipping a 10-pound package by overnight air -- the most energy-intensive delivery mode -- uses 40 percent less fuel than the average roundtrip drive to the mall. Ground shipping by truck uses just one-tenth the energy of a trip by car to the store.

In fact, each minute spent driving to the mall uses more than 20 times the energy of a minute spent shopping on the Internet. Online shopping eliminates the need for car trips and reduces congestion. Already, nearly 40 percent of people with Internet access say they go to the store or the mall less often.

The article, however, is dated 2000, and gives no sources.  It also touts just-in-time delivery.

Along with downloads, Amazon's used book network is an energy saver.  Everything I've ordered there has come via regular mail, not FedEx or UPS.