Will looking at an eye chart every day improve eyesight?

No. There are no known "excercises" that have been shown in good medical studies to permanently improve your best corrected vision. The visual system is not the same as your muscular system so more repetition does not improve your vision. Looking at an eye chart every day will allow you to test your vision and alert you to any changes or weakening of your vision.
Eye chart. No, not at all. If you memorize the letters you only fool yourself.