Same deal. After installing Yosemite, I get a pop-up every once in a while stating, "to run this web content you need to install Java Runtime" or some such. I clicked on the "More Info" button which took me to an Oracle site that states that Yosemite requires runtime version of Java 8u25 and that I shouldn't use Java 6 or Java 7. I downloaded JRE 8u25 and installed it. I'm still getting the pop-up every once in a while.
After doing some google-fu, I'm seeing that this a common problem. I'm also getting conflicting or, at least for me, confusing information:
1. Some are saying that some app on my machine is looking for Java 6, which is installed differently than Java 8, and not seeing J6 causes the pop-up. From there, I'm seeing different solutions:
a. Don't install Java 8; install Java 6 from a link at an Apple support doc. This doc is older than Yosemite's release so I'm hesitant about this.
b. The app that is looking for Java 6 needs to be updated to look for and work with Java 8; nothing I can do here but wait for updates.
c. If I create something called sym-links (or correct those that already exist, not sure which), I could make the app searching for Java 6 RE see my installation of Java 8 RE. This is over my end-user head.
d. I could install the JDK version of Java 8 and the pop-up problem goes away. I could do this but it seems excessive.
This whole problem is compounded by the fact that this pop-up shows up at different times and situations; I don't know for certain what app is causing it. Mavericks had the pop-up too but installing what Oracle told me to made the pop-up go away. It isn't working this time.