Do you lose your DEX bonus to AC while standing in the area of effect of a Grease spell?
Only if you are trying to walk
Let's unpack the situation.
You are standing around and someone cast grease on your location.
If you failed the save, you fell down. You have -4 AC against melee attacks, and +4 AC against ranged attacks, but you don't take any other penalties.
If you passed the save, you have to repeat the save every round, but otherwise you're fine. This changes if you try to move, because
A creature can walk within or through the area of grease at half normal speed with a DC 10 Balance check.
But as soon as you make the Balance check, this happens:
You are considered flat-footed while balancing, since you can’t move to avoid a blow, and thus you lose your Dexterity bonus to AC (if any). If you have 5 or more ranks in Balance, you aren’t considered flat-footed while balancing.
Because the Balance check is only called for if you try to walk, a creature that is not trying to walk does not lose Dexterity to AC. This means that you can keep your AC high by standing still.