Everywhere I read it is very cautious about oxidizing your beer. What is the worst case if your beer gets oxidized? It tastes funny?
You may get a "wet cardboard" or "sherry" flavor from oxidation, if it is bad enough. The real problem with oxidation is long-term stability. If you plan to drink your beer soon, it may not matter much. If you are going to age your beer for a while, you may experience staling reactions that will create these off-flavors.
It is important to know when you should aerate your beer (or wort) and when you should not. You want to add oxygen to the wort for yeast growth either as you put it into the fermenter, or about 24 hours after fermentation has started (with the aid of an oxygen stone). The only time that a home brewer really needs to avoid aeration (which leads to oxidation) is when bottling or kegging. (It has been widely debated in the brewing community whether or not Hot Side Aeration (the aeration of your wort while mashing, sparging and boiling) contributes to staling reactions. If you listen to the podcast linked in the comments for this answer, you will probably agree that Hot Side Aeration (HSA) is largely not a concern.)
One thing to note about long-term flavor stability, is that staling reactions are sensitive to temperature. If you have a beer that you feel has aged sufficiently, you can improve the long-term flavor stability by storing it cold. Every ten degrees of cooling (Celsius) slows these reactions by half to one-third, so a beer that may go stale in one month at room temperature, can last up to a year or more if stored in the refrigerator.
The bottom line is that it is important to avoid staling reactions due to oxidation, but there are degrees, and the primary problem is long-term flavor stability. You can mitigate these problems by avoiding aeration at bottling/kegging time, and storing the final product cold once it has aged enough.