Why does national geographic use "live curious" instead of "live curiously"? I suppose we should use adverbs to describe verbs.
In the one hand, this is very simple. There is a trend in US advertising to drop the LY adverbs and use the adjective instead. That would not be acceptable in a formal essay but it's OK in advertising as it reflects the way people talk. Play Fair, instead of play fairly. But, on the other, there are amusing things like: Think Big, which is not an example of this, since there is no bigly and Think Big is a colloquial expression.
This is an intriguing example where an ambiguous adverb serves less well than incorrect grammar.
Live curiously is correct but ambiguous, as seen at Macmillan:
curiously – adverb
- in an unusual and interesting way
- in a way that shows that you want to find out about something
Live curious incorrectly modifies a verb, live, with an adjective, curious, but is unambiguously and readily understood.
The true answer to your question, “Why does National Geographic use ‘live curious’ ...,” is probably that “be curious” doesn't satisfy high advertising standards.