As a manual/exploratory tester who works very closely with developers I find that the best way to improve my effectiveness is to learn as much as possible about programming in general, plus the languages/technologies (e.g. Drupal, WordPress) being used. Being able to code to the same standard as a developer is not a requirement to be an effective tester, but being able to hack stuff together or tweak existing code is definitely a good way to learn more about the products I'm testing and build up my overall domain knowledge.
Being able to think about a product in terms of how it might have been coded also helps me to better assess risk, discover bugs more quickly and (often, but not always) identify a likely cause to an observed problem. I also find that having deeper domain knowledge helps me to write better bug reports that are written in a format that developers can easily understand.
My advice to you (and the tester you're supporting), therefore, is to give him/her as much insight into how the software is built, for instance by explaining how key technologies work or by talking him/her through some of your code. This will hopefully help him/her to become a more investigative tester.
I wouldn't say there's anything that a developer shouldn't teach a tester, but I do think it's important that testers aren't just replicating the testing already performed by developers. Testers need to be able to think beyond the code and the software's intended functionality; instead of just checking that things are working as they should, testers should use their intuition and creativity to find edge cases and unintended functionality that a developer might never have thought of.