I think that the command axis('off') takes care of one of the problems more succinctly than changing each axis and the border separately. It still leaves the white space around the border however. Adding bbox_inches="tight" to the savefig command almost gets you there, you can see in the example below that the white space left is much smaller, but still present.
Note that newer versions of matplotlib may require bbox_inches=0 instead of the string 'tight' (via @episodeyang and @kadrach)
from numpy import random
import matplotlib.pyplot as plt
data = random.random((5,5))
img = plt.imshow(data, interpolation='nearest')
img.set_cmap('hot')
plt.axis('off')
plt.savefig("test.png", bbox_inches="tight")
