Do I need to charge sales tax for goods sold in DC if I'm a florist based in Oregon (a no sales tax state)?
In general you don't have to charge sales tax when you ship an item from your state to the customers state, unless you actually have a nexus in the customers state. In general it doesn't matter what your state tax law is, it is only based on the tax law where your locations are and do they include the state where that customer is having the product delivered.
You need to make sure that if you are using a service to sell those goods, that that doesn't trigger the nexus issue for a state. You might also have to charge a customer tax if the product/service is not being done by you, but through a 3rd party: for example a local florist is actually providing the service. Examples include teleflora and 1800flowers. The service/company should have rules about the applicability of sales tax, and how it is handled.
Keep in mind if you do have to charge sales tax, you will have research if the tax rate also has a city/county/town aspect. Some states also differ in how they handle goods verses services. So they could have set their rates such that the flowers are taxed, but the delivery charge is not.
In general, no. The legal principle is that you are only required to collect sales tax on behalf of states in which you have a "nexus". (Some states now claim that purchasing ads on blogs written in their state constitutes such a "nexus". Previously, the consensus was that the seller had to have actual business operations in a state to have a "nexus".) (But note: I am not a lawyer, and I am not responsible for the choices you make. I am also not responsible for the choices your customers make, except to the extent that I might be such a customer.)
Your customer is responsible for paying any "use tax" in their jurisdiction. Most states that charge sales tax on purchases within their jurisdiction also charge an equally high "use tax" on purchases of the same type made by their residents outside their jurisdiction. Arrangements for determination and collection of this "use tax" are made (if at all) by the customer and their government. States that have both a use tax and a state income tax tend to have a line on their income tax forms for declaring purchases subject to use tax.
As you might imagine, most people do not pay any "use tax", even if they theoretically owe it. The most common scenario under which people actually pay "use tax" is when moving from a low- or no- sales-tax state to a high sales-tax state, and the high sales-tax state makes the immigrant's initial vehicle registration subject to payment of the "use tax".
By the way, the U.S. Constitution has a provision prohibiting states from charging import duties when importing goods from other states. Since these so-called "use taxes" are de facto on the importation of goods from other states, and have no de facto relationship to how much the good is used in the state it is imported to, it is not obvious why any citizen has a constitutional duty to pay such alleged "use taxes".