In some ways, taking Bitcoin is similar to taking any other additional currency, with most of the real differences happening on the back end.
The first (and most obvious) UX change is the pricing. Do items get priced in multiple currencies? Or are they all priced in the vendor's native currency and the total presented in multiple currencies at payment time? Does the user set their currency somehow and it just carries through?
If pricing everything in multiple currencies, then the system has to be clear in differentiating the various prices. It's important here that there's no chance of the user becoming confused as to the cost of an item.
At payment time it's again important to insure that the user knows what currency is in play, and what their payment options are for that currency. Then there's the issue of the Bitcoin payment flow - it's nothing like a CC transaction. You'll generate an address to send the Bitcoins to, then you'll have to wait for the user to send them. There will need to be timeouts, etc to make sure that the user doesn't sit around for days before completing the transaction.
Once you've taken a payment in Bitcoin there's further user experience issues with making sure you've actually got the payment before you ship product - confirmation email timing vs. receiving appropriate confirmations of the transaction, vs when to send the order to the warehouse to ship, etc, all play into the total transaction experience. The process needs to keep the user informed while making sure that the vendor doesn't get shafted.
As to the vendor side of things: Bitcoin is a whole other currency (a highly volatile one) which presents similar accounting issues to any business that deals in multiple currencies.
Among other things, if the vendor is paying for product in native currency, then taking payment in Bitcoins, it's important that the vendor have some good way to deal with the exchange rates.
Whether to change the prices on the fly is part of that, but also the vendor has to decide when to move money between currencies and whether to try to make a little extra by carefully timing their transactions (playing the market), or whether to simply move all Bitcoin to native currency ASAP, so that they lessen their exposure.