Based on these two bug reports, it appears that Taleo and some other sites rely on a previous nonstandard behavior in Firefox. Now that Firefox uses a more standard behavior, their scripts fail because they also rely on a nonstandard behavior in Webkit-based browsers. The Firefox developers are not yet convinced to switch back to the nonstandard behavior, that is, they think the sites should update their scripts:

   843075 – Firefox 19 fails to display Sudoku grids (regression)
   845882 – Oracle Taleo job search engine doesn't work since Firefox 19 

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Source: https://support.mozilla.org/en-US/questions/951236

As it says, TALEO seems to be using non-standard scripts that are avoided in modern browsers, that's not a problem of Firefox, a problem of TALEO, sorry.

Taleo sucks. Everybody knows that, (except for these companies who use it, apparently). Anyway, we don't have any control over what Taleo does or whether the companies to which we want to apply use Taleo. The only thing we can do is switch to another browser like Internet Exploiter, which is slow and annoying to try to use. That's why we prefer to use Firefox over IE! Why give us a reason to dislike Firefox when the solution is to just fix it and to stop waiting for Taleo to fix theirs? It's been months, and nothing's changed!