oDesk automatically pauses contracts after there has been no activity for a month or so.
I have a number of these. I leave them because the work on the contracts in intermittent but if you don't expect any work to come in from your paused contracts you can always send them an email asking if they want to close the contract.
Hmm... good question.
You're really not making any money off of a paused contract. And oDesk obviously doesn't like seeing contracts go on and on without any work being done, so that's why they pause the contract automatically.
So a paused contract which isn't earning you any money is not really to your advantage. But closing a contract yourself is not optimal, because that can be held against you by oDesk's algorithms, and if there are many such instances your account can be put under review.
There IS an advantage to having an open contract if you do work once in a while for a client. That client does not need to hire somebody else, and when they need a little something done, they simply ask you to do it.
There IS an advantage to having the contract closed: You don't draw any suspicion from oDesk for having open contracts that don't earn money, and you gain an additional job that is shown on your profile. An additional job in your job count and an additional feedback score and hopefully a nice comment from your client.
So for me the decision comes down to how likely will I be paid by this client in the immediate future to do some additional work. If likely, keep the contract open.
If it is unlikely, then send a VERY SHORT note to your client, thanking him for the opportunity to work for him in the past, state that you're available if he needs you in the future, but that because there is no work right now, could he, as a favor to you, close the contract and leave whatever feedback he feels is appropriate.
Don't pester the client after sending one note. Wait a couple weeks. Then if you really want to close the contract, do so yourself, and hope that the client leaves feedback when prompted to by the oDesk automated interface.