To understand how these two sports might conflict, we need to delve deeper into how muscles are controlled.
Chad Waterbury had this nice picture showing how a motor unit touches several muscle fibres. I suggest those who don't know what motor units are start reading up a little bit :-)
Taking the biceps as an example, this muscle has about 200 of these motor units to stimulate all the muscle fibers.
But you want to contract your muscle, they don't all go fire at the same time. This is due to the size principle, motor units of the smallest size will be recruited first. For a more detailed explanation, I can recommend Human Physiology: The Mechanism of Body Function
As this picture shows, your muscle fibers are divided into roughly three groups. First the green ones are recruited: these are your Type I fibers, meant for endurance. Then when you start to recruit more motor units (by increasing the force of the contraction), you recruit the yellow ones: these are your faster Type II fibers, but as you can see they aren't the strongest muscle fibers. If you would contract even harder, you would start recruiting all your motor units (well as much as possible) and you reach your maximal power output.
What does this have to do with shooting?
If you lift a lot of heavy weights, you're mostly training that last group of muscle fibers. When you do this enough, you're muscles start to adapt to become better at it. However, my hypothesis would be that this influences how your muscles recruit muscle fibers.
Now if keeping a gun stretched out requires 20% of your biceps forces, that means it has to recruit the yellow muscle fibers. But since they work in an on/off kind of way, they'll recruit a surplus of fibers and thus produce more force than required, so you have to counter it with your triceps. But that muscle is also a lot stronger, so your muscles have a very hard time at load balancing to keep your arm steady!
Furthermore, weight lifters aren't trained for fine-grained motion, so regardless of whether my hypothesis is entirely correct, you're training your muscles for the wrong kind of exercise. So yes, I'd say they are somewhat conflicting