Leaving out the accent marks seldom produces confusion, but it may pose a problem in some cases. Sí ("yes") vs. si ("if") is one notorious case. The diaeresis over ü is definitely not a problem. I'll be hard-pressed to think of a word that could conceivably be misread due to its missing a diaeresis over u.
So I wonder, is there a pair of Spanish words that only differ in a diaeresis over u, so that the diaeresis is needed to tell them apart?