Would you tell me if I need to use not or no in the context below?
The box has to be not/no taller than 72 inches to fit into the trailer.
If both are possible, is there any difference in meaning?
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"No taller than" is probably more common (at least in American English) in this context, but they're both possible.
The strange thing is that, at least in my opinion, "no __er than" generally emphasizes that the two things being compared are close. For example, when you say, "Andrew is no smarter than Bruce," it suggests to me that Andrew might be almost as smart or as smart as Bruce, but in any case no smarter. "Andrew isn't smarter than Bruce," is on the other hand, is probably neutral on the point of closeness of comparison. "Andrew is not smarter than Bruce," (with actual emphasis on the not), would be appropriate if you were strongly countering a claim that Andrew is, in fact, smarter than Bruce.
However, the effect is pretty subtle, and in your example, this connotation of "closeness" through the use of "no" doesn't apply at all. The box "has to be no taller than 72 inches" sounds perfectly natural and doesn't imply that the box is expected to be almost 72 inches. Whether the missing connotation is due to the context, or possibly due to the fact that your construction doesn't allow for "isn't", or due to something else, is hard to say.