I will be flying into Heathrow and will be staying in a flat between Victoria Station and Pimlico Station by Warwick Square. Should we take the tube to our lodging or another mode of transportation? Also, we will be purchasing an oyster card, but will be flying out of Edinburgh. Where can I turn in my card? Thanks.
You don't have to turn in your card. The cards are transferable, so you could give it to a friend who was visiting London, or just throw it away. The card has no personal data on it unless you have registered it. Try to ensure that there is minimum credit on it before you leave London, and put the lost credit down to one of the costs of travel.
You say both "I" and "we", so have you considered the National Rail travelcard, which gives you discounted entry to various London attractions? There are pros and cons, but the subject is debated here at length fairly regularly.
How are you leaving London? You only mention how you are leaving the UK, by flying from Edinburgh.
Don't throw your oyster card away, its worth £5!!! Just turn it in at the last possible tube stop you will be using in London, make sure it has staffed booths, and get your 5 quid refunded back to you along with any balance you have on the card. To get from Heathrow to your hotel, take the Piccadilly line from Heathrow (where you will buy your oyster card) then transfer at the Hammersmith stop by merely walking about 25 feet to the other side of the platform to catch the District/Circle line to Victoria station. Ask your hotel if its walkable from Victoria. They should be able to recommend your next step.
As others have said, you don't have to turn in your Oyster card when you leave. I bought one years ago and kept it, as work was taking me through fairly frequently, and I have always enjoyed London.
As to taking the Tube to your hotel or not, the Tube is the most economical way to get from LHR into the city. If you aren't carrying copious amounts of luggage, that is how I'd go. You are approximately halfway between the two stations.