I have a method called ImageTag set up in my Page which takes an image object as a parameter, and returns a string.

public function ImageTag($image) {
...
}

However, when I try to use this from my template, it doesn't work:

$ImageTag($MyImage)

I get an error saying "Trying to get property of non-object". My guess is that this is because I'm not actually passing the image object to the method, but rather the rendered image (i.e. a string with an image HTML tag). How do I access the actual image object from the template to pass that as a parameter to the method?

Might this be what you're after?
http://api.silverstripe.org/3.1/class-Image.html#_getTag

Actually, I think that's the opposite is what I'm after. I need the pass in the actual Image object to my method, not the HTML tag, which is what's passed in at the moment.

This method is on the Image class. You can use this instead of your method to get the tag for the object.

Ah, I see your point. But I needed a custom image tag, so the default one will not do, I'm afraid.

The best solution is to create an Extension & apply it to Image, and define your getCustomTag() method there.
Then in your template, you can simply call $MyImage.CustomTag.

Yup, that's how I ended up solving it. I'm just curious if it's possible to solve the original problem - i.e. passing the actual image object to the method.

No, because it's already been converted to a string by Image's forTemplate method, as you discovered:
http://api.silverstripe.org/3.1/class-Image.html#_forTemplate

You could pass the ID to the method and Image::get()->byID($id) if you needed the object back in the controller.
Though at that point, you'd be better off appending the data to the object before it hits the template.

Extension solution is best, as you know.