Can a virtual object exist without an observer? I'm a beginner in optics and my understanding of virtual objects is that a virtual object is defined as a point from where incident light rays apparently seem to be coming from.
But can a virtual object exist outside of the human brain? Can it exist without an observer? I also read that real image of a vitual object can be produced. Is this true? If so please explain this paradoxical (seemingly) situation.
 A: Don't get stuck at the word "virtual", it has nothing to do with the image not being a well defined thing in the physical world. It is just (arbitrary) nomenclature.
A real image is one that can be projected onto a screen (that is, where the light coming from one point of the object is approximately collected in one point in space).
In a virtual image the light propagates as if there was an object or real image at the position of the virtual image, but if you put a screen at the position of the virtual image you cannot see it there (because, for example, in the case of virtual image due to a concave lens the virtual image is before the lens and the light ray have not been deflected yet).
This also tells you that you can form a real image from a virtual image, since the light rays behave as if there was an object at the position of the virtual image, so we can place a lens which forms a real image of the virtual image.
Actually, exactly that happens when a human sees a virtual image (which happens, for example, when using a magnifying glass), the lens in the eye projects a real image of the virtual image onto the retina.
A: A real image is a very tangible thing. Make a small real image of the Sun using a magnifying glass, put a match where the image is and it will burn.
As for virtual objects, disregard lenses and just consider a mirror. Are there objects on the other side of a mirror? No, but it’s easier to calculate light paths by assuming virtual objects emitted the light, because the consequences of physical laws are such that light rays reach an observer in exactly the same way as they would if the objects had their virtual positions and shapes without any mirrors or lenses in the way.
Also consider the problem of calculating the path of a billiard ball. In many cases the problem is much simplified by applying rules of optics and pretending a virtual cue hit a virtual ball from outside the table. It's not a philosophical question whether the virtual cue exists, of course it doesn't, but it's a useful practical method of solving problems.
A: According to wiki:

In optics, a virtual image is an image formed when the outgoing rays from a point on an object always diverge. The image appears to be located at the point of apparent divergence. Because the rays never really converge, a virtual image cannot be projected onto a screen. In diagrams of optical systems, virtual rays are conventionally represented by dotted lines. 

In other words,the image formed on the screen reaching your eye,that appear to arise from an actual object,however there is no such object present at apparent source.The best example of the virtual image would be image formed by plane mirror.
You can indeed produce a real image from a virtual object like Sebastian stated.
Like take for an example a plane mirror.
For a plane mirror to convert an virtual object to real image ,the light rays coming from the object has to converge behind the mirror so as to produce a real image of it by using another mirror or lens.

