How can you remove the UNITY logo from the exported game?
If you're talking about the Unity splash screen, then you need to update to Unity Pro. If you mean the application icon, then you need to set an icon in the Player preferences. To do that in Unity 3, go to Edit->Project Settings->Player and select the stand-alone settings tab. Click the subsection that says "Icon" and select an icon for each slot. The high-res one can be up to 256x256 (I think).
Can't remember how to do it in Unity 2.6, but it should be fairly similar.
Buy the pro version o.o
Maybe its still in the trial, but you better contact unity directly for that.