We are selling our house and the tank is half full (approx. 250). Who owns this?

You own the propane but its likely not something you can remove. You can try to negotiate a price with the buyers of your property. Around here that is worth about $450 IF I had to buy it. See what the buyer will offer. You can also call your supplier to see if they have any additional offers.

You own the propane but its likely not something you can remove. ...

I think that, if nothing is said in the sales contract, it may be implicit that the propane comes with the tank. It would be best to put it in writing that the tank is about half full and that seller doesn't warrant it to be full, but that what is there goes with the house. Then negotiate the price on that basis.

Items like propane and fuel oil in tanks belong to the person who paid for them ( assuming they are paid for)

If selling a home in the listing you will have to specify that you want it paid for, but in general, expect the buyer to merely offer a price for the home including it.

It is a larger deal, if you are renting and have to deal with a land lord moving in or moving out.