My suggestion would be to get rid of the drop down, as the user cannot change options as they please. Moving from "open" to "closed" is a one-time action, and cannot be reversed.
I recommend that the current status is just shown as a label (with colour if you wish). If the current status is set to "open", then show a link that allows the user to close the current request - make it as clear as possible what the link does.
Something like this:
The reason I have opted to use a link, rather than a button, is because the action is expected to be less common. Therefore, using a link makes the function more subtle, rather than having a big button that is just begging to be clicked.
Of course, as this is an irreversible action, you need to ensure that you prompt for confirmation when the user clicks the link. This can be a simple yes/no dialog box. If you want the user to be extra sure they are agreeing to close it, then you could add an additional checkbox to the confirmation dialog.
The user is then forced to "acknowledge" the action before they can confirm (as the button is initially disabled). This added step effectively forces the user to stop and think about what they are doing - as opposed to just being able to click buttons without reading the message.
I would argue it is probably overkill for this particular feature, and is better server for actions such as permanently deleting something. However, I wanted to give you the option if you think it is required or not. You could just have the dialog without the checkbox.