I've asked a couple (OK. A few) questions that have been closed. I've noticed that they still seem to gather additional "views". Can closed questions still earn badges? And, no, I'm not sure if I tagged this question correctly.
Yes, closed questions behave normally except for one thing: you can't post an answer to a closed question. You can still earn badges for votes or views on a closed question, or an answer to a closed question.
Questions that are closed for reasons other than being a duplicate are normally deleted after a while. When a question is deleted, you stop earning any badge for it, but any badge already earned is not taken away.
For more information, see How do “badges” work? on the main meta site.