Some philosophers argue against truth bivalency, and say that not every statement must be true or false, but some statements can be untrue without being false, or truth-ambiguous, or both true and false (I can link to articles proposing these positions, though I've personally never understood that last one.) This comes up most often in contexts of the liar paradox and its many iterations.
Would such a position throw a wrench in making truth tables? Would it only be possible to make three-tiered truth tables? I know that the consequences of rejecting bivalency for the law of the excluded middle is discussed, but I haven't seen anyone talk about (truth-false) truth tables - perhaps because nobody uses them seriously, or because the answer is obvious and I just haven't figured it out.