The locus classicus for the tetralemma in Buddhist philosophy is Nagarjuna's Mulamadhyamakakarika. While a strong formulation of the tetralemma does not in fact appear in the work, it's clearly what he's driving at.

My view is that if you look at how Nagarjuna argues in the MMK, it''s clear he must have held to the law of the excluded middle. His argument is reductio ad absurdum (Prasangika). Nagarjuna sets out to refute the concept of inherent existence. To do so he establishes that if you try to claim that phenomena inherently exist you will be forced to defend contradictory and absurd statements. If there were no law of the excluded middle, this would not be a problem - one would feel free to contradict one's self.

Countless illustrations of his reliance on the law of the excluded middle can be found in his work. I openned MMK to a random page and immediately found:

"An existent and nonexistent agent
Does not perform an existent and nonexistent action.
Existence and nonexistence cannot pertain to the same thing.
For how could they exist together? "
- MMK VIII, 7

It's clear that Nagarjuna at least regarded the law of the excluded middle as a core principle of reason. And every formulation of the tetralemma in Buddhist philosophy is ultimately based on Nagarjuna. I regard him as one of the two or three most important philosophers in world history, along with Aristotle and perhaps Kant.

The question then becomes, what does the tetralemma mean, given that he accepts the law of the excluded middle? For a thorough answer to this question I would refer you to Tom Tillemans's outstanding essay "Is Buddhist Logic Non-classical or Deviant?". His conclusion, which I quite agree with, is that the tetralemma does not mean:

P is true
P is false
P is both true and false
P is neither true nor false


What it actually means is:

X does not exist such that P is true
X does not exist such that P is false
X does not exist such that P is both true and false
X does not exist such that P is neither true nor false

This is a classical (i.e. non-contradictory) logical argument. In this case "X" is "a phenomenon that exists inherently", i.e. without depending on anything else. The tetralemma expresses the general form of his argument - that if one asserts inherently-existent objects, one will be forced to defend contradictory conclusions.