Many of the observatories on Mauna Kea open their domes before the sun has set. It's not because they're observing anything, it's to help equalize the temperature within the dome to that of the outside air. Having said that, many observatories can observe during the day, especially those that operate in the mid-infrared (beyond about a wavelength of 2.5 microns) and the submillimetre (such as the JCMT and CSO). Many years ago we started the commissioning process of an instrument called Michelle by observing after the night shift had ended until around nine or ten in the morning. After that the sun is high in the sky and the risk of the primary mirror concentrating sunlight on part of the dome becomes a real risk. Even before then we would regularly go out into the dome to make sure we weren't setting fire to the observatory!
Those were long and tiring shifts, I remember them well.
During the day heat builds up inside the domes. Some observatories actively control the dome temperature via air conditioning whereas we (UKIRT) control the primary mirror temperature to keep it as cold as the expected night-time air. This is a project we've been doing for a couple of years now and it's made a significant improvement to the delivered image quality in the first few hours of the night.
No matter how these things are handled though, it's always best to try and ventilate the dome for as long as you can before observing starts as any difference in the temperatures inside the dome and the outside air result in local turbulence and reduced image quality. Gemini, one of the observatories that employs active cooling of the dome, will still open the vents early when they can, as seen above. I think the only issue I'd have with this is that sunlight is apparently being allowed to strike the telescope structure directly. Our own experience with this is that it's enough to heat the structure so that thermal expansion occurs. It's only a few microns but it makes a difference. On the other hand, Gemini is a much larger telescope and therefore has a much larger thermal mass than us, so it may not be a problem.
At UKIRT we open the dome ventilation system as soon as the night crew are at the summit although the vents facing the setting sun remain closed. The dome slit is opened as well to help with the cooling process although we make sure the slit is facing away from the sun! We rely on the summit winds to blow through the dome and equalize the temperatures although that's a bit of a problem when it's calm. Fortunately, that's doesn't happen very often.
This is also the time that the summit tourists find their best spots to view the western sky for a sunset of a lifetime. I tend to join them when I'm on the summit because no matter how many times I've watched a Mauna Kea sunset, they simply never become boring.