Could OLED displays be made to produce a voltage when off?
As Ignacio points out in the comments:

Most large screens (if not all) are LCD types, usually either TN (Twisted Nmatic) or IPS (In Plane Switching), with a certain type of backlight. "LED TV" just means they are using LEDs to shine on the back of the display. Some more expensive ones might use 16 different sections, or even more with independently controlled LEDs, to give you the experience of more range in the brightness. In other words, faking the static contract ratio, by having one section that contains a lot of black dim the LED, and another that is very bright turn it all the way up.

But that's about the level of sophistication in the LEDs that "drive" the main picture. OLEDs are expensive and take a lot of time to make, whereas LCD is a piece of cake. Remember several Mobile phone brands making an OLED model around 2012-ish and then quickly announcing they'd also start using normal LCDs due to sourcing issues? I do!

To answer your actual question, assuming there is an LED-Pixel TV, with say 4K resolution. That's 8.29 million LEDs with each an R, G and B.

((I'd guesstimate its price tag somewhere in the $ 50000+ range))

And then a LED of that microscopic size will have never have the same effect/performance as a normal size LED that they put in a 3mm housing. 

So, what does a LED do when light falls on it? It tries to induce a current, because some of the photons hit the substrate such that electrons get annoyed in one general direction, much like in a photo-diode. (&lt;-- concise slang for a whole lot of techno gibber about valence, energy states, recombination, reverse action, transport, electrons and electron gaps)

That's like in a Solar cell! Yes, exactly, but also no. The difference is in the magnitude of the effect. In a Solar Cell huge amounts of electrons get buggered into moving, seriously, busloads of busloads. That's partly because of their structure, but also because of their giant surface compared to a measly diode. In a photo diode, it'll be one or more orders of magnitude less per unit surface area, because they are still engineered to do that, but also to be as linear as possible, this is sort of a trade-off. Solar cells are efficient, but far from linear in their response.

In a LED the materials are composed and ordered such that as many electrons flowing cause photons to be generated as possible (what else would you engineer it for?). Unfortunately this isn't the way to go if you want to generate energy. A LED will only induce a few micro ampere when you shine a light on it. And that even also depends on its colour. A Red LED can't see blue very well, for example.

So, even if we assume a microscopic Organically grown LED can induce 1uA and reach 0.1V (I think this would still be very generous) while that current flows, a TV with 8.29 million sets of three LEDs would only reach 8.29 * 10^6 * 3 * (1 * 10^-6)A * 0.1V = (8.29 * 3 * 0.1)W = 2.49W

Would you engineer the electronics to capture that into a TV that already cost you way too much in OLED materials? Let alone, mount your personal TV in direct light for that? It'll basically mean that if you mount your TV in the sunlight, it might be able to stay on stand-by for free. Might be able to.