Many photographers consider the UV filter a necessary accessory for a new lens, and while it does have some useful qualities, there are a few situations in which you’d be better off without one.
1) When you’re using other filters
Beginners to filtration and photography tend to think that because the UV is a clear filter, you can stack a second or even third filter on top of it without affecting your exposures. This is not the case. The general rule is to have as few pieces of glass in front of your lens as possible, for optimal results. Each element will reduce transmission and cause some light to scatter slightly, resulting in a softer, darker image. The addition of multiple circular elements in front of the lens may also cause unwanted vignetting.
2) When shooting into the sun or bright lights
UV filters are a common cause of lens flare, and while some really well-designed filters may not always create this issue, cheap filters and lights in the frame rarely mix well.
3) When you need the best image quality
Lenses are designed without filters, so they will theoretically function at their peak potential without one. Not to say that using a specialty filter will not give you amazing results you couldn’t achieve otherwise, such as a polarizer or 10-stop ND. But UV filters don’t do much when it comes to digital photography. Cameras are not as susceptible to the effects of UV light as film is.
4) When you’re worried about lens damage
This tip straddles both sides of the fence of the UV filter debate. Many people use UV filters primarily for protection; however, certain kinds of lenses can be damaged if the front element makes contact (the convex Canon 24-70mm f/2.8L II comes to mind). Also, if you accidentally shatter a filter, shards of glass could potentially scratch the attached lens.
A great resource for information on this topic is the B&H UV Filter Buying Guide, which explains what ultraviolet light is, why UV filters are so important for film cameras, the role that UV filters play in digital photography, and differences between various kinds of glass.