Would a visible light beamforming "telescope" be feasible? If you had a phased array of high speed photodiodes that were spaced sufficiently far apart, would it be possible to apply beamforming theory to create a steerable visible light "telescope"?

Traditional, earth-bound telescopes are limited by the size of their primary mirror due to the mechanical properties of figured glass as the size increases.  Additionally, the larger the primary mirror the more robust the tracking mechanism needs to be to handle the weight.

It seems that if you could instead "virtually" track the object by beamforming the output of a horizontally mounted photodiode array, you are no longer limited by the size of a primary mirror and/or infrastructure to physically move the telescope.  Angular resolution could be increased simply by making the array larger.  Furthermore, your tracking accuracy is only dependent on your software processing and electronics, not moving parts.

Of course, the problem likely becomes dealing with immense quantity of data collected from the photodiode array, and the sheer physical size of the array.  

Light travels at about 299,700 km/s in air, so ~29.97 cm/µs.  That seems within the realm of high-speed electronics given a large enough array (10x10m?)

Difficult, huge amount of data...but it doesn't seem impossible unless I'm misunderstanding how beamforming works?
Not in the way you think, no. 

Digital Beamforming works because the digital receiver is able to sense the phase of the incident radio wave; that phase is retained.

Photodiodes don't sense the phase – they only sense the intensity. 

Also, this requires the whole incident radio or light wave to be phase-coherent across its front – that's the case for laser light, but not for two random photons emitted from a huge fusion fireball, I'm afraid. So, the second requirement also falls flat. Generally, even non-laser sources of light have a coherency period – but the more different wavelengths contribute the smaller that gets. 

In TX, that principle works – if you feed a laser through an array of digital phase modulators (optical crystals with electrically adjustable refractive indices), you can build phase-coherent multiemitters. 

In RX, that might also work, if the received coherent light can be made to fall through a microscope objective, and then onto an acousto-optic deflector, you might, via modulating that deflector's state, be able to build a selective/phase-shifting summer, leading to beamforming. But: that's particle physics sized electronics normally, i.e. suitable for situations where you have a very strong laser illuminating very small points (in fact, I learned about AODs just today, and they're usually used the other way around: take one beam of laser, split it into multiple parallel beams, focus these, and trap quantums in the focus point. Cool stuff.).

Anyway, there's also mathematical accuracy limits to digital beamforming, and I'd must admit that I don't know whether they'd even allow higher precision with the kind of timing accuracy we can do today. Then again, phyisicists invent the coolest devices, and maybe you can