The only item that MUST be changed to go from a triple to a double crankset is the crankset, itself. Things that MAY have to be changed are the BB, FD and RD. Things that probably SHOULD be changed are the chain and cassette.
The BB may need to be changed if you cannot adjust it enough to work with the new double crankset. For instance, with a BB that uses the internal nuts you can adjust the BB left or right with these nuts. Or, with something like a Phil Wood BB, you can slide the spindle through the press-fit bearings to work with either a triple or double. If these adjustments don't work, you will need a new BB.
Same with the FD. A triple can be adjusted to work with a double. You will simply have to put up with the additional travel that you don't need to move the chain the required distance needed with a triple. But, FDs are relatively inexpensive, so I would think this is a part that you would go for. But you generally don't HAVE to do so.
The RD from a triple will work just fine with a double. I used the long cage RD with both. Shifting is a bit slower due to the long length of the cage, but works just fine with either. If I was going to race, I would opt for the short cage RD to get the quicker shifts. But I don't, so the long cage that works with the triple works just fine with the double.
Depending on the mileage and condition of your chain and cassette you may wish to change them to start out with everything new. If opt for using your old you may find that they wear out your chainrings on your new double faster than a new set would. Also, you may need to remove a few links from the chain should you decide you want the new RD with short cage. However, if you stay with the triple RD you will not need to remove any links. Most likely your triple is a 30/42/53 and your double will be a 39/53, or close to this. The length of the chain depends on your big chainring, so they are the same or within a tooth of each other, so the chain should work fine.
If it were me and I knew that I was going to stay with the double, I would replace everything with what I wanted for a double and sell the triple parts as a group on eBay. If in decent condition you will probably get about 40-50% of the cost of the new components, assuming you are buying the same quality level. For instance, selling a Centaur (or Ultegra) triple and buying a Centaur (or Ultegra) double.