Which is stronger: a polar covalent bond or a nonpolar covalent bond?

It's complicated, but in water, polar covalent bonds tend to be weaker.

The true answer to the question is that it's complicated. For instance, an O-H bond is weaker than a C-H bond because the shared electrons of the bond are drawn closer to the O, leaving the H easier to remove in a polar such as water. So, in this aqueous case the polar bond is weaker. (This is reflected in a higher Ka for hydrogens of alcohols than of aliphatic hydrogens)

However, a bond table will tell you that O-H has a higher bond enthalpy than C-H, and this is true in a vacuum where water is not assisting in the H removal. This can be attributed partly to differences in atomic size. Smaller size results in a shorter, stronger bond - regardless of polarity. F-H therefore has an even higher bond enthalpy, but in water the H is even easier to remove.

So, the long answer is it's complicated; but, in a polar solvent such as water, polar covalent bonds tend to be weaker. This is a good enough assumption for most biology classes.