What happens to the projection if the object is put right at the focal
point of the lens? The image will be projected at infinity, right? How about if you place the object even closer to the lens, i.e. between the lens and the focal point?

 

Think in terms of collimation. This is basic optics physics, I think. On either side of the focal point the image will either be inverted (and spread) or just spread out.

From the title of your post, I thought you were talking about a Frensel Lense.

 

What happens to the projection if the object is put right at the focal
point of the lens? The image will be projected at infinity, right? How about if you place the object even closer to the lens, i.e. between the lens and the focal point?

If the object is between the focal point and the lens then the image is virtual and magnified. This is how a magnifying glass works. You can't project a virtual image on a screen. Google should find you more details.

 

Mind you no lens in perfect so no focal point a singularity and that objects are 3D so infinitely projected without dispersion impossible.