I've been doing some Tabata thrusters recently. I have been close but still slightly below my max heart rate (HR). I suspect I have been at a higher HR while playing some sports (eg: after 6 full courts sprints when playing intramural soccer), but did not measure HR then.

My question is: what happens if you reach max HR? Do you faint? Do you die? Or does your body slow you down somehow (cramps/low blood pressure?). A few times when playing sports my body slowed my down. Vision became blurry and I just could not move for a few seconds. Would be curious to know what happens then.

you get tired and slow down

If you are healthy you get tired and slow down. That also applies if you are in general good health and your MAX HR is kept lower by beta blockers (taken for hypertension or glucoma) but, a person with heart disease could have a heart attack. It is not something to worry about unless you are older, out of shape, and haven't exercised in a long time. But, some people have had heart attacks in the doctor's office when taking a heart stress test.