I would say Tones and HenryT are right. It is the length of the vibrating body that matters.
This is quite noticeable (and listenable) with big organs. If you take a look at the organ on my avatar you'll see the façade is spread in height. Some pipes sound at the top of the organ, others are near to ground level. In this particular organ the difference can be as large as 15 meters.
Now, as you all know, heat travels upwards. When a church is full of people, heat rapidly concentrates at the top of the church. In consequence the pipes dilate and go out of tune - they get lower in pitch.
Is it noticeable? It most definitely is. But as each department of the organ tends to be used by itself, differences are not disturbing.
Now when the upper department of an organ has both wooden and metal pipes, that can get worse. Wooden pipes just don't change their pitch with temperature (their volume does not increase) whereas metal ones do - their pitch lowers. And there is where you get problems.
With central heating these days, the problem has been either solved or made much worse - it all depends on the air flow.