If you want to drop the 2 most significant bits, then you don't have to do any bit shifting.
byte a = analogRead(...);
Gives you the 8 least significant bits.
byte a = analogRead(...)>>2;
Gives you the 8 most significant bits.
say you have this:
If you assign that to a byte it simply truncates any information from bit8 up, so you are left with:
Now if instead you want the upper bits, you would bitshift by two:
However, I don't really understand why you want to do that. All it would mean is that you end up with the result going from 0-255 four times over one full rotation of the pot.
If you want to reduce precision, you want to drop the lower bits. If there is dead space on the pot it means simply that 8bit isn't sufficient precision for what you want. Trying to drop the upper bits essentially just reduces the usable area of your pot to 1/4 of what it was.