He has misunderstood Feynman's point. Pi does not change, though the ratio of diameter (D) to circumference (C) of what you determine to be a circle might.
Pi is equal to the sum of the infinite series 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 ...
This obviously does not change under any physical circumstances, however the ratio of C to D can depend on the topology of spacetime. Imagine the Earth was completely smooth and spherical. If you drew a small circle (say 1m diameter) on the surface and measured C and D, C/D would be a pretty good estimate of pi, but draw circles with diameters of 1, 10 and 20 km and you begin to see a growing discrepancy where C/D is progressively less than pi.
Fairly obviously this is because you are drawing your circle not on the infinite flat plane of geometers but on a sphere. The same is true of the universe, the overall topology of spacetime may be positively curved (spherical), negatively curved (saddle shaped) or flat, and this will subtly effect certain geometric relationships over intergalactic distances. Famously it determines the sum of the internal angles of a "triangle", >180 for positive curvature, <180 for negative and exactly =180 if flat. This fact, combined with some very clever measurements of anisotropies in the cosmic microwave background is strong evidence that the universe is very very nearly, if not actually, flat.
But this is only the overall topology, spacetime is also distorted on a much smaller scale, albeit to a much lesser degree, so just like time runs that little bit slower on the surface of the sun, so do circles and triangles behave just the tiniest bit differently.
