This is kind of a silly question. A complete answer would fill up a book, In the context of a board design the answer is irrelevant (I'll explain later). But it's still a useful question.
Follow the data-sheet. Especially notable areas, if the designer has decided to tell you NOT to ramp the voltage on first power up too fast, don't get creative and decide you know better. Even if it is a hassle to design a power supply that ramps slowly. Most indications given will be subtle, if the data-sheet is strident, then really pay attention.
There are many failure mechanism in ICs but the impact on the chip subsequently requires intimate knowledge of the internal workings. You can have a failed transistor and it may not ever be noticed or it may take out the whole chip. Even the designer may not know, FMEA is performed but usually only in key areas (FMEA = Failure Mode Effects Analysis).
A lot of the constrains on the chip are imposed by the process. The foundry dictates design practises, placement etc. and they have software that checks for violations. Other limitations are operational and these translate directly into Maximum voltages, heat sinking requirements, dV/dt on signals etc.
Because of the increased cost, the complexity of the design and the knock-on effects of getting it wrong (schedule delays etc.) for the most part most chip designs will have most of the bases covered. But there are notable exceptions when the process failed.
So, asking about chip design failure modes with respect to board design is kind of like opening up your car hood, removing a random bolt and asking "will this stop the car from working?" if you remove the oil pan plug - yes. if it one of several redundant bolts on the valve covers - no. Where the real answer is, "don't be removing bolts!".
For some designs, the chip designer will sometimes put "gotcha's" into the design that if the board designer is clever will cause them to pay attention. The best interactions is when the board guy says "why did you put that signal out on that pin over there, it messes up the blah blah etc." these are almost always the start of wonderful conversations and usually a long time interaction and friendship. The response back is usually " very good, how ever, this is the signal flow in the chip, and if you did it that way ...." the response is always a " Eureka".
An IC will not over current unless you have damaged it or are operating it outside of it's specifications. Over-current = bad things already happening, which only makes sense. A chip will only present a certain sized load to a voltage source, this means a fixed current. Saying that over current is what kills chips is like saying a fixed voltage source driving a fixed resistor might have a failure because the resistor might go over current. The over current is a fault/effect it is NOT a cause.