You can turn the axes off by following the advice in Veedrac's comment (linking to here) with one small modification.
Rather than using plt.axis('off') you should use ax.axis('off') where ax is a matplotlib.axes object. To do this for your code you simple need to add axarr[0,0].axis('off') and so on for each of your subplots.
The code below shows the result (I've removed the prune_matrix part because I don't have access to that function, in the future please submit fully working code.)
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
import matplotlib.cm as cm
img = mpimg.imread("stewie.jpg")
f, axarr = plt.subplots(2, 2)
axarr[0,0].imshow(img, cmap = cm.Greys_r)
axarr[0,0].set_title("Rank = 512")
axarr[0,0].axis('off')
axarr[0,1].imshow(img, cmap = cm.Greys_r)
axarr[0,1].set_title("Rank = %s" % 128)
axarr[0,1].axis('off')
axarr[1,0].imshow(img, cmap = cm.Greys_r)
axarr[1,0].set_title("Rank = %s" % 32)
axarr[1,0].axis('off')
axarr[1,1].imshow(img, cmap = cm.Greys_r)
axarr[1,1].set_title("Rank = %s" % 16)
axarr[1,1].axis('off')
plt.show()
Note: To turn off only the x or y axis you can use set_visible() e.g.:
axarr[0,0].xaxis.set_visible(False) # Hide only x axis