The first thing I did was try to find out just how big Altair is. Surprisingly, it's not that big. According to Wikipedia its radius is 1.7 times that of the Sun. (Granted that this is Wikipedia, but as I can't think of anyone who would gain political advantage from Altair being a particular size we can provisionally trust what it says.) Also, I'm assuming that refers to the radius at the equator. So, on the right side of the image is a picture of the Sun, to scale. That's not too shabby -- looks like CHARA could take a pretty decent photo of the Sun at 16 light years' distance.
We're already doing well here, so I decided to keep going. In front of the Sun, you can see the four popular gas giant planets in our own Solar System, Jupiter, Saturn, Uranus, and Neptune, also to scale. It's a little hairy, but looks like you might be able to get a pixel's resolution on the interferometer and pick up Jupiter, Saturn, or a larger super-Jovian object; at least, it would then depend on just how bright the planet was (in the infrared -- that's the wavelength the original photo was taken in) and if you could blot out the light from the central star to a great enough degree, areas I'll confess I'm ignorant on.
Oh, and that tiiiiiiiiny blue-white dot? Just off the limb of the Sun, below Jupiter? Yeah, that's Earth to scale. I don't think even CHARA is picking that up, not even if we pointed it at Alpha Centauri (four times closer.) Perhaps someday, but it's not going to happen just yet.