Yes, a simple actionscript using flash will help you. You can create a document with a button (the button will be the door, you can fill up the background of the button with the door image). Your flash movie may contain 2 keyframes (1 with the default background including the image-button and another keyframe with the image to be displayed when button-pressed). Add some actionscript for the button (basically a GoToAndPlay(# of keyframe with the image) and also a "stop()" action on the image keyframe. Here is a basic example on YouTube Click here. By the kind of thing I guess you want, a little bit of flash animation will make it look better i.e. (door opening clip, 2nd image slide, etc)
As Chris said, a simple JavaScript image swapping code will work too. Here are a couple of examples in StackOverFlow Click Here.
Also you can try some CSS, for example a DIV with the background door and a Hover action that sprite the background to the calendar (click would not be necessary, just hover). Here is a video
I hope any of those solution will work for you, I recommend the CSS one, because it can work with mobile devices, don't require javascript to be enabled, and it's fast and easy.