Is the phrase "results show" appropriate in academic writing? I once heard of a professor saying that it is not appropriate to say that "results show", because it is not the results that actually "show" anything, and a more appropriate way of putting it would be that the results "support" the idea of something. 

I am not sure if this is overthinking or there is actually a point made.
The professor is suggesting that in academic writing one needs to be precise and one also tends to be more formal. Things that are natural in conversation, for example, may not be the best way to express something in writing - especially academic writing. 

But it is easy, and natural, to slip up since we normally spend more of our time speaking and writing informally than formally. 

But yes, most "results" only give evidence that something is true. Only in some fields do the "show" or "prove" something (say, mathematics). 

I think the point is valid - precision and formality are likely good, but condemnation of such terminology is probably overblown. In most cases the reader will make the right assumptions about what is meant. Novices might be misled, however, if claims seem a bit overblown. And intentionally overstating a conclusion is wrong in general.