There a lot of reasons that tractor can start fine and then die. For example, if it runs long enough to warm up then dies, yet starts after it cools, it's likely it's the condenser. The one I'm going to zero in on, seems to be the most common though, and that is that it's just not getting enough fuel. Even this can cause you to have to look multiple places. A blockage can be in the tank, the fuel line or even the carb.
I'd just finished rebuilding my Allis Chalmers D14, and after the initial requisite half hour warm up and oil change, suddenly it started dying. It was running perfectly and still starting perfectly, except, after it ran for about 15 seconds, it would die. Pulling on the choke would keep it running a few seconds more. Leave it sit for a minute, and the process would repeat it itself... run fine, then die. I was pretty sure I had a fuel blockage. Since we answer questions about this constantly on the forums, I thought I'd give a little background into how to find it.

First of all, why would there be a blockage. Well, these are old tractors, that shiny unpainted steel in the tank builds a lot of scaly rust above the fuel level, virtually any time the tractor sits. Many of the machines we work on haven't seen a full tank for a long time, and worse, the gas may have evaporated long ago, leaving only dried varnish and rust. If you cleaned the varnish up, you may have gasoline flow for a bit, but as you fill the tank, that scale all gets knocked off and tries to head right on down to the carb. The photo to the left shows just how much scale broke loose in the supposedly clean tank I was working with. When this happens, finding where it decides to stop is the task to get the machine running again. Most often, it will be in the sediment bowl.
This particular tractor hadn't run for 12 years and I'd already noted and cleaned out the varnish from the sediment bowl, well, more accurately I noted the varnish, cleaned it a bit and then decided to swap the bowl with a new one I had the foresight to pick up, knowing what I was going to see. If you don't have a replacement handy, it's a good practice to pull of the sediment bowl before you actually complete your other repairs and take a look down the stem of it. When you open the valve, you should have a straight path for fuel to flow, if you don't, you have gunk or scale that's going to hamper your first attempt to get your tractor running. But varnish like this won't be the problem if you have a tractor you normally use and suddenly starts exhibiting these symptoms. The problem will be rust scale.
I should note, all that scale shown here was not the problem, the mass of scale shown in the photo had already been cleaned from the tank before even attempting to start the tractor. Usually rust in the system takes far less to cause problems than that shown here, like one or two small chunks of rust are enough to ruin your day.
When this problem started, I troubleshot it by removing the secondary fuel filter and checking the flow at that point. Many don't like using fuel filters since it duplicates the job of a sediment bowl, but I do. The reason in this case was two fold. The tractor always had one on it, and I would have to get an original one piece fuel line if I didn't use one... which I had one extra, but... the more important reason, is that I had just spent more than I'm willing to admit turning the engine into a virtually new engine, from new crank and rebuild kit, to governor, distributor... as I say, virtually a new engine. I was uncharacteristically meticulous about the rebuild, and the thought of even minute rust particles finding their way through the carb and into the intake, past the valves and into the cylinder, made me cringe just a bit. A modern fuel filter catches the stuff you can't see, and often you don't have the luxury of fixing the biggest source of contaminants coming into your fresh new system, the tank itself (unless you seal the tank, but that is another story, I chose not to with this tank). A sediment bowl is great at stopping the big stuff, but not the little stuff. If you keep the tank filled, eventually it will clean itself out, but during the time it is, it's nice to have the extra protection of the fuel filter.
Back to the troubleshooting, when I removed the fuel filter, I'd effectively cut the system in half and would know whether to work back toward the tank, or toward the carb. By the way, at this point I shook the fuel filter intake into a paper towel and as expected, there was a brown hue of rust on the towel, stuff that didn't make it's way to the engine, more evidence of the value of a modern fuel filter, but not the problem. When turning on the valve at the sediment bowl, only a trickle came out. It was almost a stream, but not really. If you think about it, the smallest orifice in the system is at the sediment bowl, it's the place where the gas dumps into the bowl, and the two tiny outlets are 2 roughly 1/8 inch squares. Not being a math expert, I'll just guess I should have as much gas flowing out as would fit unrestricted through a 3/16s inch ID tube. That's actually a lot of flow, should be a strong stream. Anything less, means you are going to starve the engine of fuel. Dripping out, definitely won't cut it.
At this point, I was certain that the problem was from the fuel filter back, probably at the sediment bowl or tank outlet. Since I'd put a new sediment bowl on, it ruled out the bowl's screen and I'd also already took a welding rod and poked into the tank outlet from above to break up any rust that might be clogging the outlet into the sediment bowl. The problem could only be in the line between the sediment bowl and the fuel filter, or in the sediment bowl inlet itself. After replacing the fuel filter, I removed the sediment bowl, and turned on the valve with coffee can held underneath. Still a dribble rather than a nice flowing stream. It was in the sediment bowl inlet. Having already cleaned some scale out of the bowl, I could be reasonably sure a chunk that could fit through those little nearly-8th inch openings in the inlet, had come down into the inlet and was catching other chunks and slowing the flow. Rather than remove the unit from the tank, I took a small pointed instrument (a dental pick) I keep around for such things, and probed up under the small cap that blocks the end of the inlet. Sure enough, the large chunk broke up and immediately I had a nice stream of fuel.
The "small cap" I mentioned, is what I presume the designers of the sediment bowl used to keep fuel from creating a continuous turbulence in the bowl when in operation. Instead of the fuel dumping directly downward, this cap forces the fuel to shoot out the sides, allowing the particles at the bottom of the bowl to remain where they are and not be suspended and sticking to the screen while the engine is running. I show a photo of one that's missing that cap and one that isn't. Also note the screen on the one that has the cap, it's beyond service, yet I just pulled it out of a running machine. Since that cap is a choke point, it's a great place to look for the blockage. It actually can get a little more complex, because not only is that "cap" a choke point, but also the jog the casting takes inside the sediment bowl. If you look directly into the hole at the top of it, the inlet, you'll not see daylight even though the outlet that dumps into the bowl itself is also in the middle of the casting. They designed it with a jog. That is a great place for large chunks of rust to hang up and start a log jam. Either from the top (if you remove the unit from the tank or use a rod like I do), or from the bottom (if you pull the bowl off), you can break it loose.
After removing the blockage, the tractor started and ran fine again. My problem was fixed, but what if it hadn't been there. If you look at the photo of the larger bowl with bad screen, had I been using it, those chunks of rust could have made their way through the line and right into the carb. Large chunks like that, won't go into the engine. They will stop at the carb and probably give the same symptoms. Some carbs have a screen at their inlets and the problem will be solved by cleaning that. Others don't and if the chunks are the "right" size, they will make their way right into the carb, and though likely be caught in the small passageways or float valve, you will be looking at a carb teardown and cleaning to fix the problem. That's another reason why I like a real fuel filter inline.
A few loose ends I'll tie up. I mentioned using a welding rod. I mean a nice long steel rod that you'd use with an oxyacetylene torch. I poke that into the hole in the tank which is actually the sediment bowl inlet. It may do the trick, in this case it didn't. The blockage was at the very bottom of the inlet. I mentioned sealing the tank. The correct way would probably be to seal the tank after cleaning it so you never run into this problem. I cleaned my tank well, used alcohol to attempt to get moisture out of it and then kept it taped while I was finishing the rest of the tractor. The problem was that the completion took me several months longer than expected. I have a sealed tank that has lasted 20 years and doesn't appear to be a problem, but I've always held that as a last resort, usually reserved for tanks that have gotten so bad, they have pinholes. Still, it would be better with a restoration to make certain all parts of the system are perfect. Keeping an old tractor tank full is the last thing I wanted to expound on. That helps keep rust down and keeps less air from reaching the fuel, especially with ethanol, the fuel will last longer.
One last thing is another preventative measure you can take. The sediment bowl in the last photo has a screen on it. This is a screen for early Fords that can be adapted to other sediment bowls. It's obviously going to keep the large sediment out of your system. The downside is that the fit requires using a sealer to keep it in place that is impervious to gasoline. I should have put one of these on my sediment bowl.