To the answer above, I will add that if the lenses are cheap and not designed for multi-day wear without interruption, then the next day the image quality in reality may no longer be HD.
I once had to spend the night not at home, and I didn't have the lens solution with me. In the morning there was a desire to remove the eyes from the sockets and rinse them thoroughly. In one eye there was a cloudy veil, the other saw a little better, but also not perfect, it was an hour's journey to the house, and all this time was torment for me. When I removed the lenses, naturally, it passed.
I can answer as a person who slept in contact lenses - no, they will not.
Yes, and there is no reason to believe that such effects are possible, lenses only change the optical power of the eye, no more - to the image in a dream they definitely cannot directly influence.