How to make a tap faucet close automatically?

Yes, you can add a spring to the internals of your cheap faucet to make it close automatically. You can use the self closing tap springs for Intertap faucets, but you will have to cut them down to fit. You should also consider grinding the cut ends smooth to avoid scratching the insides of your taps. I did this and it worked for me. Of course, finding a spring that is manufactured to the appropriate length, diameter, and strength and from a food-safe material would be preferred. I do not know the size and strength required.

I purchased self closing tap springs for Intertap faucets with the hope that they would work in my faucets. I purchased my keezer used and cannot find identifying marks of any kind on the faucets, so I don't know what brand they are. They are chrome plated, not stainless, so I assume a basic "economy" faucet that came with a keezer kit purchased by a previous owner. The Intertap springs were WAY too long. After some trial and error I cut them down to about 1/4 of their original length and ground the cut end smooth. They now fit and appear to close the tap. I don't have any beverages in the system right now, so further testing to come.

UPDATE: I have been using the keezer for about one year with the springs in place. With the "standard" black plastic tap handles the springs close the taps very effectively. With larger, heavier tap handles the springs still work, but, as you would expect, there is less "snap". I always give the taps an extra small push just to make sure they are completely closed.

Update: I've made my standard taps self closing with the S-1098 Compression Spring from Century Spring.

It's a stainless steel spring with the following measurements:

Force: 102lbs/inch (179 N/cm)
Free length: .560"
Solid height: .350"
Outer Diameter: .453"
Inner Diameter: .325"

This strength adds a fair bit of resistance to pulling the tap open, but it's not a problem. The tap closes quickly and cleanly (just a couple drips, about like a normal hand-closing) on its own after being let go, provided the spring is seated well in between the faucet and shaft. Sometimes it takes a couple tries to sit properly. I just set it in and attach it like normal.

The one question that remains is if the springs will hold up (I've installed two in beer taps and one in sparkling water). While they are made of stainless steel, they are not polished, so I will report back in a while with how they continue to perform.

Research below

Okay, I've found a spring that roughly works, but is not of an acceptable material (and sadly I don't know it's exact strength because it was in a sampler pack).

The packaged spring C-632 (7/16 x 1-1/16 x .041) from Century Spring felt pretty responsive when it was alone in the tap (quick clean closing action) but didn't close all the way. In the sample pack I found one that is about the same length and slightly stronger tension (12mm x 28mm x 1.25mm) and it closes nearly as cleanly as just shutting off the tap by hand, and solves the problem of leaks when the door slams shut.

The problem is that carbonated water sitting on it for a day or two led to a lot of corrosion. Both of the springs were probably too long for this purpose (I measured the inner length to be about .35" when open and .45" when closed) and likely not a food safe material. I've ordered a handful of stainless springs and will update the answer when I find a suitable size.