Why can't websites access JRE in Yosemite?
As suggested by Moto_Nomad, this appears to have been caused by the Facebook Video Chat plugin requesting Java Runtime Environment. I needed to undertake the following steps to locate and disable this process.

Launch Terminal, and type the command launchctl list | grep facebook

This results in something like the following:

-   1   com.facebook.videochat.&lt;yourname&gt;.updater
-   0   com.apple.facebook.xpc


Where &lt;yourname&gt; is your Facebook id. To delete the process, type the following command:

launchctl remove com.facebook.videochat.&lt;yourname&gt;.updater

Remember to replace &lt;yourname&gt; with the Facebook ID returned in the first step. To check the command has worked, type launchctl list | grep facebook again. This time you should see the following:

-   0   com.apple.facebook.xpc


Having done this myself I can confirm that the dialog shown in the question is no longer appearing.