I've been reassembling a friend's old BBQ and one of the three burners seems to be leaking gas where it connects with the gas intake. When I turn it on, there's a flame and I can see/smell something burning inside, so I turn it off immediately. Anyone know why, and how to fix?
It looks like the burner tube is not sliding back far enough onto the gas outlet on the control valve. Then some gas leaks there instead of going into the burner. All the gas should go into the burner tube. The gas will aspirate air into the the tube, mix with the air and give blue flames at the little holes. If there is orange flame, the adjusting collar on the tube needs to be rotated. Your problem should be a simple fix of sliding the burner tube tight against the valve. It is a pretty forgiving system: I have a 20 year old grill that is on about the 5th set of burners, not made for the specific grill and have not had a problem. (I use the grill a lot). At the end of season I buy whatever burners are on sale, they have all worked so far.
This is obviously not safe. It looks like there is a leak in your valve. I'd have to see photos of the valve taken apart, but this could be caused by rust or just not putting it all together properly. In addition to water and water vapor (and plain old oxygen), heat will cause oxidation of the metal, so you may need to replace the valve assembly.
If it is simply a fitting that's leaking, you can try using pipe joint compound to properly fit those pieces together with no leaks. See also Blacksmith's answer as it seems like he's onto something that I'm not seeing in the photo.
Do you smell gas when the burner is turned off? If so, a little soapy water may tell you where the leak is.
If you want to supply additional photos I may be able to offer more advice.
Yellow flames mean that the gas it not mixed with enough air before combusting. If I understand correctly, there is a nozzle that injects gas into the tube-like burner. Around the nozzle there is probably a small gap to allow the gas to mix with air (like the bunsen burner you may remember from high school).
When the flame ignites in the video (around 5 seconds in), you only see blue flames along the first few centimeters of the burner. I suspect it might be clogged at that point, causing the gas to flow back out of the air intake.