If I were to set out a training plan for the beep test, burpees would definitely be in the s and c section.
I think the individual athlete will need to approach training slightly differently depending on his particular muscle make up and background.
The beep test tests the limits of both the aerobic system and the anaerobic system at the same time, so we need ideally to prepare both systems.
At no point will the athlete be running his fastest, he will eventually fail because he will be overcome by the by-products of the anaerobic system (which he will progressively rely on more and more as the test progresses). We therefore will also do well to increase the athlete's max speed ability so that he is using a smaller fraction of his full capacity, thus drawing less hard on his anaerobic system early in the test.
So we want to;
increase anaerobic power (ATP delivery) and anaerobic buffering (against hydrogen ions),
increase the body's ability to clear the by products of the anaerobic system
increase the aerobic system's own ability to deliver ATP to take the heat off the anaerobic system,
increase the athlete's pure speed ability (so his muscle usage is more fractional)
Whatever we do, it should be to these goals and we would do well to try to tick all these boxes with some component of the training.
First off, probably a universal supplementation recommendation that will pass a drug's test would be;
Beta alanine daily @ 4-6 g for at least a month before the test (this is to max the last gasp power of the anaerobic system),
Caffeine on the day 1 hour before the test @ roughly 6mg/kg (experiment to find your personal sweet spot) -this is to max the oxygen delivery of the aerobic system (it does change the fuel of the aerobic system a bit too but the test is too short for this to really count),
Baking soda about 2 hours before the test @ 0.3g/kg (with a couple of practises first so you don't crap yourself) -this is to buffer the muscles and nerves against the hydrogen ions released with the lactic acid that the anaerobic system turns out.
The individual athlete should ideally look to his own muscle makeup in choosing his training plan. If he has a higher proportion of fast twitch fibres, he needs to be very careful about overloading his muscles with lactate and causing damage that will reduce his gains. If he is a good sprinter, footballer, soccer player, or lifter, he probably has more and better developed fast twitch fibres. If he comes from an endurance background, he can be a little more free with lactate baths (this is a bit counter intuitive and missed by a lot of coaches) .
What is not always appreciated is that muscle fibres work in harmony in endurance events. The very favourite food of slow twitch fibres is actually lactate. So if you have a lot of well developed slow twitch fibres full of hungry mitochondria, fed oxygen by a great blood supply, they hoover up your lactate. You can take a harder work out than the guy who does lots of gym work and has a less well developed "waste recycling system" Smashing yourself in the gym might be good for hypertrophy but we are looking for performance here and the fast twitch guy needs to be gentle on himself or he cooks himself early and (against his expectations) lowers his performance gains.
Both types of athlete are in a situation where their main strength is also their weakness when it comes to the beep test (this is what makes it such a cool test, it's a bit of an equaliser). The endurance guy has a weaker anaerobic system that is soon running close to capacity but which can be cleared fast. The well trained fast twitch guy can't clear himself fast but is using only a small fraction of his power and has more in reserve at the mid point of the test.
So it will be a bit personal how the athlete approaches the training and both types should look to their weaknesses if they have already been working their strengths in their main sport but some ideas are;
10 x 200 metres off 2 min rests to develop the anaerobic system and speed endurance without cooking yourself too much
4 x 400 metres off 3 mins
Running hard into and out of a turn with a 1 min rest at each end (to practise the form, deceleration and acceleration)
A weekly zone 2 long run to develop mitochondria and slow twitch muscles (this will be counter intuitively faster for the endurance athlete and must be very slow for the speed athlete or he will be filling up on the hydrogen ions that come with his lactate)
Vo2 max intervals -run @ 2 mile race pace for X time. Rest for X time, repeat quite a few times. (Hard practise to use sparingly but it develops max oxygen delivery).
The odd longer harder effort at about 10 k pace for about 20 minutes to practise lactate clearance or a fartlek type session where you vary pace between jog and sprint.
Pure speed development -run almost as fast as you can for no more than 10 seconds and only 6 at full speed, (aiming to tax only the Creatine phosphate energy system and the nervous system, keeping your lactate minimal). Concentrate on taking very fast steps with high knee lift. Rest at least 3 mins between reps to allow creatine phosphate to replenish.
Plenty of good S and C -the kind of things the marines put recruits through.
And last but far from least -practise the test itself seriously once a month and gently maybe once a fortnight. Bear in mind this is a maximal test. Don't do this on a bad day. Training is not the main event. You don't have to take it to the limit to still get the benefit of the training. Probably you will actually get more physical benefit by a sub maximal practise -your gains will be retained more because you will not have cooked yourself as much. The occasional maximal practise will be good mental preparation.