first of all, why would they? secondly, PETA!

That is cruelty to animals, and just a really dumb idea.

Did you ever see that episode of thats incredible where they dropped the worlds biggest ball of elastics down into the grand canyon, if you did then you'll know why.

because an elephant is too heavy for the spaceship.

I know, I know, I'm mad for attempting to answer this question seriously, but here goes. Let's assume that, rather than any space administration such as NASA is cruel to animals and forces them into going into space, and instead go with the notion that the elephant volunteers for the astronaut space program. Let us also assume that, rather than failing the medical with stupendously high blood pressure and obesity, it passes. Let us also assume that, rather than looking like it has been shrinkwrapped, it is in fact given a space suit that fits. Let us also assume that somehow it can breath in really, really strongly and force itself to fit through the shuttle doors to allow entry. Let us finally assume it it called Ernie, because it seems disrespectful to keep referring to this incredibly brave individual as "it". Firstly, Ernie will weigh a lot more than your typical astronaut. When you see the size of the main fuel tank and additional booster rockets that are necessary to propel the relatively small and light shuttle into orbit, it gives you an idea of just how much propellant is required to make the trip into space, escaping the gravitational pull of the Earth. With Ernie's extra weight, unless he follows the same diet as Courtney Cox for six months (i.e. half a stick of celery a week), he's going to have a significant effect on the amount of fuel required, and so additional tank will be needed. His next problem is fitting through the airlock, but since we have somehow dealt with the weight problem already posed by magic, we can solve this conundrum by magic as well. Or we can just stuff him in the cargo bay pre-launch, in which case, Ernie, by the time the shuttle reaches orbit, is now Ernie, R.I.P. I'm fairly certain no safety belts are fitted in the cargo bay for pachydermal passengers. So, now we have a dead elephant ready to be jettisoned from the cargo bay. Great, but what will happen once he is free from the space craft. Well, since the space craft is travelling at 17,500 mph to maintain orbit, this also means that Ernie (deceased) is also travelling at 17,500 mph. Which means he is travelling sufficiently fast to also keep orbit. The Earth's gravity is pulling on him, just as it is pulling on the shuttle. In fact, when the shuttle is in orbit, the astronauts aren't floating because of no gravity. They are actually under the influence of about 91% of gravity they are on Earth. The difference is that they are in free fall - the Earth's gravity pulls the shuttle towards it but it is also going fast enough so that as the shuttle drops towards the Earth, it's momentum moves it over the "horizon", so it in effect falls past the edge of the Earth. In essence it is continually falling towards, but around the Earth. In the same way, Ernie will now also be falling around the Earth in orbit. So his corpse wouldn't actually fall to Earth, it would maintain orbit around it instead.