Laptop speakers make noise when audio jack is plugged in

The solution to this is to disable Auto-Mute Mode in alsamixer. Detailed instructions below.

Open up terminal and start alsamixer:

alsamixer  

You should see a handful of tall columns with labels underneath, something like: Master, Headphone, Speaker, PCM, Mic, Mic Boost, S/PDIF, S/PDIF Default PC, Beep, Auto-Mute Mode, Internal Mic Boost

If you don't see these categories (specifically, if you don't see an "Auto-Mute Mode"), then try selecting a different sound card with F6. In my case I had to select HDA Intel PCH.

Notice "Enabled" above Auto-Mute Mode. Navigate to that item with the left/right arrow keys, and set it to Disabled with the up/down arrow keys. This should solve your problem, and you can exit alsamixer with Esc or by closing the terminal window.