If you mean simply "is not visible" then very little, we'd have a second of darkness, not unlike an eclipse, and then we'd be back to normal. If on the other hand, you mean "ceases to exist momentarily" then you're talking about something else entirely.
The orbits of all of the planets and a great number of other objects in our solar system are the result of a constant acceleration towards the sun, under the influence of it's gravity.
The Earth, for example orbits the sun at an average of velocity about 30 kilometres per second (depending on how far out it is, as the orbit is elliptical, rather than circular)
By removing the sun, and as a result it's gravitational effects upon other objects in the solar system, then rather than continuing to accelerate around a circular path, they will instead continue to travel in a straight line at a constant speed instead.
This might not seem like much, initially, but as every planet, comet, asteroid and spec of dust in orbit around the solar system would move independently of each other, their orbital paths would change once the sun resumed normal service.
Some objects would travel closer to others, and others would travel further away. This would result in many objects finding themselves in unstable orbits where the slight increase in gravity between them could give them a small kick, which over time could lead to objects being ejected from the solar system entirely, or tugged out of their orbits and into other objects and planets.
On a larger scale, the removal of the sun would also remove its protective magnetic field from around our solar system. This field, known as the heliosheath, protects us from a lot of radiation and particles from outside our solar system by deflecting them away from us.
By removing the heliosheath for a second, we'd be allowing all of this radiation into our solar system. Even at the speed of light, this radiation wouldn't travel far, after all it takes around eight minutes for light to reach us from the sun, and the outer limits of our heliosphere appear to be around 84 times further away, but once restored, the suns magnetic field would be far less able to safely deflect all of that radiation away from us, and much more radiation could potentially reach us.
This radiation would lead to some very interesting aurora in the atmosphere, and the interaction with Earths own magnetic field could cause problems for orbiting satellites and ground-based electrical grids, much like solar flares do today.
Beyond this, I'm not sure, but I think it'd be an interesting experience to live through.
Image: Shutterstock/K13 ART
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