Any such functionality is dependent on the player or viewer used. I know of no image formats that require any kind of interaction from the viewer, but some video formats do (or, rather, did) use it.
In addition to Quicktime mentioned by 0xea, the Windows Media/ASF format allows embedding of script commands, and one of them is opening an URL:
http://support.microsoft.com/kb/828026
When a content owner creates an audio or a video stream, that content owner can add script commands (such as URL script commands and custom script commands) that are embedded in the stream. When the stream is played back, the script commands can trigger events in an embedded player program, or they can start your Web browser and then connect to a particular Web page. This behavior is by design.
However, this behavior:
- relies on the player supporting it (e.g. I don't think VLC or mplayer support it)
- can be disabled by the user
So, I would not rely on it.
EDIT
Speaking of images, I've just realized that SVG is using XML. XML files may contain references to extenal URLs, such as DTDs or schemas. Additionally, SVG itself supports referring to other elements in the same document or sometimes external ones:
http://www.w3.org/TR/SVG/struct.html#Head
I've just tried it and opening the following SVG results in a fetch to the remote server in Opera, but not IE:
<?xml version="1.0" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
<svg width="8cm" height="3cm" xmlns="http://www.w3.org/2000/svg" version="1.1">
<desc>Local URI references within ancestor's 'defs' element.</desc>
<defs>
<linearGradient id="Gradient01">
<stop offset="20%" stop-color="#39F" />
</linearGradient>
</defs>
<!-- reference to a local definition -->
<rect x="1cm" y="1cm" width="6cm" height="1cm"
fill="url(#Gradient01)" />
<!-- this element triggers a fetch of the specified URL in Opera -->
<rect x="1cm" y="1cm" width="6cm" height="1cm"
fill="url(http://example.com/test1)" />
</svg>
So, once again it's viewer-specific but you probably have more chance of this working.