I have never tried to include the source directly inside the .tex file but in case your trouble is with the metapost code it may help to start with a known-good stand-alone source.
I did this recently, so I suspect that it still works (although I have edited out some stuff I think is not needed so do let me know if it fails). Here is the input file. It draws the Venn diagram for the union.
% MetaPost input file with chapter one pictures.
outputtemplate := "%j-%2c.mps";
% Label the sets (from pp 29 pf MetaPost manual)
picture pa, pb;
pa = thelabel(btex \tiny $A$ etex, (.9venn_circle_left_shift,1.15venn_circle_top_shift));
pb = thelabel(btex \tiny $B$ etex, (1.1venn_circle_right_shift,1.15venn_circle_top_shift));
unfill bbox pa;
unfill bbox pb;
At a (Linux) command line I ran the four commands
mpost set.mp followed by
tex mproof set-00.mps followed by
dvips -Pwww -omproof.ps mproof finally followed by
this last pops up a viewer for the figure.
To include it in my doc I used this.
This is inside Beamer. I don't see that I did anything special to get the graphicx system to recognize the .mps ending but I may have forgot the details; let me know.
(Edit: someone changed what I wrote and in particular dropped the necessary initial paragraph. I added something like it back.)