I don't see how it is "outlandish" that a master martial artist, who has Mastered Death, can temporarily resist the heat of lava, or dodge and parry the blades of a Blade Barrier.I don't think "exploit", or the OP's term "broken", is a good way to describe Mastery of Death. But it does have potential for unforeseen, and possibly undesired, consequences.
The first potential for wonkiness is along the lines of things other people have brought up. When the monk knows he can't die, it enables outlandish behavior like swimming in lava or dancing in a blade barrier. Many people won't see this as being any more of a problem than abstract hit points and other game mechanics. But I think Master of Death takes a few more chips away at verisimilitude (for those that care about it).
Isn't the point of character abilities to be used by players to affect the game? I can't see how the things you describe here could be problems. I mean, as Gygax described way back in his DMG a dragon can have the 100 hp fighter chained to the cliff, ready to breathe, and the fighter can survive that fiery breath with a lucky saving throw: or three breaths from an adult red dragon with three successful saves (32 hp per successful save).Mastery of Death has some interesting effects on resource management. If managing hit points and healing is a factor in your game, this changes the game a lot. If you know an enemy has the ability to hit hard, it suddenly becomes better to tank with a 1 HP monk than a 100 HP warrior. After all, Ki can be completely replenished after a short rest, but HP takes a long rest, and HD takes two days. Again, lots of people probably won't care. But for those that do it can result in significant changes to tactics.
One of the complaints some people have about 5e in general is that it's too easy to not die (especially compared to early editions). This ability just exacerbates that problem. And, IMNSHO, the ability to not care about death just screws with the story sometimes. Does the bad guy have me pinned to a wall with a gun against my head, demanding I tell him my secrets or die? Who cares? By 13th level he'll have to reload his six-shooter twice before I have to make a death save. Have you ever had to choose which friend to save with only seconds to keep them alive? Well, your Death monk friend is guaranteed to last an extra minute even if he's unconscious.