Every time I talk about Lightroom I am asked some variation of this question. I usually answer it by saying "I don't know.... Do you?" Of course I quickly follow up with a short explanation of why someone would or would not need Photoshop in addition to Lightroom.
Lightroom is for photographers. Photoshop is for graphic artists and designers.
OK, put down the torches and pitchforks while I explain!
Lightroom was designed from end to end with a photographer's needs in mind. The tools available in the modules all address some aspect of the photographer's workflow. Photoshop, on the other hand, was not designed with the photographer in mind but with the graphic artist in mind. It's a powerful and rich toolset but only a small portion address a photographer's needs. It is arguably the premiere pixel based editor but do you need all that power?
Lightroom will meet the needs of 85% to 90% of photographers 90% to 95% of the time. This is especially true now that plugins like Color Efex Pro, Silver Efex Pro, (Nik Software), Photomatix Pro (HDRSoft), Noise Ninja (PictureCode), and others are available directly in Lightroom, without the need for Photoshop. Now, more and more of a photographer's needs are met directly inside Lightroom.
Since version 2, Lightroom has offered localized adjustments and masks. Even though layers aren't technically available in Lightroom, the touch up tools and adjustment brushes mimic many aspects of layers. All this combined with digital asset management, slide shows, website creation, an excellent print engine, and export/upload capabilities, make the purchase of Photoshop harder and harder to justify.
If you need to use layers, complicated masks, blending modes, alternate color modes (CMYK or Lab), 3D, or other things that are well beyond the reach of Lightroom then you may need Photoshop. Keep in mind that the cost and the learning curve for Photoshop are both pretty steep! I recommend that you really look at your toolset in Lightroom to make sure you really can't achieve the outcome you want before going down the Photoshop road.
For the most part everything I've said here about Lightroom is equally applicable to Aperture. While tools and terminology differ the decision process is the same.
So do you need Photoshop if you have Lightroom?
I don't know..... Do you?