The role of atomic theory in biology
Yes it is still true for all of biology, of course physics has expanded a bit, and likewise we have discovered cases of biology exploiting these new parts, for instance photosynthesis might involve quantum tunneling. 

Feynman's quote basically restating that there is no magic or supernatural aspect to biology, no vital essence or spirit required to explain how it works or seperate living matter from non-living, it is all just chemistry and physics. The most complex chemical process ever discovered but still just chemistry and physics. Of course he does not mention chemistry becasue of course chemistry is just physics as well, the laws of physics control both, so saying it is a result of atoms obeying the laws of physics expresses both.  

He is saying biology is still obeys the laws of physics and are made of atoms, this is true for all of biology, but as Feynman points out took some time to be accepted, keep in mind this is the time of the Miller–Urey experiment which showed organic molecules could form through entirely non-biotic means, which was not as widely accepted at the time as many think, especially to a general audience. Many similar discoveries a t the time were showing the same thing, there was nothing special about biological molecules or their interaction. things like evolution is the inevitable result of the interaction of basic properties (reproduction and heredity)just as a salt crystal is the inevitable result of the interaction of charge of sodium and chlorine. Vitalism still hung around in the popular mind and a few corners of academia at the time Feynman was growing up so taking a few moments to help dispel this notion in his lectures makes a lot of sense, especially given how much advancement was done by abandoning it.