What would happen if the fatty acids in a cell membrane were polar molecules?

Fatty acids in the cells are amphiphile, it means they have a polar side (phosphate group) and apolar side (carbon tails).
The polar property helps the cell to adapt with the aqueous medium (in both sides of the membrane).

The apolar property helps the membrane to keep its integrity. Since they "hate water" the apolar side of phospholipid will stay on the inner side of the membrane and be joined together to keep the integrity of the membrane. It contributes also to the viscosity of the membrane.
If this apolar property doesn't exist, The membrane will be too water soluble and dissolves in it, and the cell could not keep its integrity

The arrangements of lipids in the membrane makes the cell semi permeable .