In general, nothing. Well, they’ll heat up like anything dark does under sunlight, but they won’t *do* anything, and the extra heat should be well within the panel’s ability to dissipate. There’s no completed circuit, so there’s either no flow of electrons, or extremely little flow. It isn’t wise to leave the thing disconnected for a *very* long time, like weeks, but a day or two would almost certainly not do anything meaningful to a typical solar panel.
Now, if it’s a relatively fragile type of panel, it’s possible that the extra heat could cause some damage. Further, in general, there’s really no *need* to leave it disconnected–it doesn’t cause any harm to let it run, after all, it’s not like you’re draining battery or the like. If you’re genuinely just not going to *use* the solar panel at all for an extended period (maybe it’s a cabin you visit only some of the time?), then it would be wise to cover it or store it away (if it isn’t fixed in place, that is) just to be very safe. But leaving it completely unplugged for a little while shouldn’t really do anything to it.
A black object that is exposed to the sun will convert very close to 100% of the light that hits it into heat. This is why black objects get very hot when they’re sitting in the sun during the summer, when the air isn’t cold enough to cool them off.
A solar panel is basically a black object that works by converting ~18% of the light that hits it into electricity instead of heat. If the solar panel isn’t connected to anything, then that ~18% of sunlight that would have been converted into electricity is instead converted into heat. At that point, your solar panel is basically indistinguishable from any other inert black object that has been left in the sun.
To be more specific, a solar panel works because the sunlight hitting the panel pushes electrons in the front of the panel through a sort of one way switch that only allows the electrons to move to the back of the panel. Once in the back of the panel, those electrons leave, pass through a transformer, and then re-enter the top of the panel. This creates a constant flow of electrons from front->back->outside the panel->back inside the panel->front->repeat.
If you disconnect the solar panel, you’re cutting the wire that leads to the transformer. Because the electrons in the back of the panel have nowhere to go, the electrons in the front of the panel can’t move to the back. Since the electrons in the front can’t move to the back, there is no movement of electrons in the panel and no current is generated. The sunlight is still being absorbed though, so it just heats the entire panel up like it would any other black object.
One thing that hasn’t been mentioned in the other (excellent) comments is that the voltage across the two wires will be a good bit higher than the typical rated voltage of the panel. You will see in the specifications of a panel that there is a max power voltage and an open circuit voltage. Basically the electrons are still being pushed by the sunlight and so charge builds up on one side.
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