Basically, the camera move will affect how easy or difficult this will be. Also, whether there are any moving objects that cross over the object that you wish to remove.
Basically, what you have to do is recreate whatever is behind the object that you want to remove. The easiest way to do that is if you have really good photography of the location without the object in the scene that you wish to remove. You can then use this to make a patch that covers the object to be removed, grade it to match, match the camera movement so that it sticks, and match the grain stock so that it doesnt look stuck on.
If you have an object that crosses over this patch, you may need to extract it so that it can be brough back on top. The usual method would be rotoscoping, but sometimes you can get a chromakey or lumakey, it depends on the situation.
If you have a camera move that would require parallax, it becomes much harder, but can still be done, i can explain further if anyone is remotely interested past this stage ;)
Of course, all of this depends on your ability to use some higher end compositing softwares, which can be hugely time consuming in itself to learn.
I did this work full time for a couple of years, and it happens all the time in films, especially higher budget films that can afford all the vfx work. I've had shots where 2/3rds of the frame has had to be replaced in this manner, and it can get INCREDIBLY complex, but the basics aren't too painful.
Hope it helps...
Nick