I just got a new bathtub installed and there is a crack in it. The bathtub company is replacing it, but it is now leaking on to the ceiling below it. Are there any suggestions on how I can prevent the leak from getting worse other than, "don't use the tub?" Also, is the bathtub company responsible for any damage caused by the leak? Thanks.

Don't use the tub is the only fix for this. And no, I don't think the tub company will volunteer paying for any water damage. You can ask, but I highly doubt it. Try contacting your home insurance agent and see if the water damage is covered. It should be. If you don't have insurance, then don't use the tub period.

Don't use the tub at all, because if you do have insurance that would pay for the damage or if the company where the tub was bought might fix the damage, neither will pay for repair of any damage that you caused after you realized that the tub was causing damage.

Also if you do have insurance your deductable will probably be so high that it will be more than what the damage will cost to repair.

Some water damage can run into the megabucks category. Been there, done that. I had insurance to pay for a water pipe bursting and having to redo a LOT of damaged flooring.

I just got a new bathtub installed and there is a crack in it. The bathtub company is replacing it, but it is now leaking on to the ceiling below it. Are there any suggestions on how I can prevent the leak from getting worse other than, "don't use the tub?" Also, is the bathtub company responsible for any damage caused by the leak? Thanks.
This should not be your problem to repair.
Follow the advice already given and don't use the tub until the leak's repaired. If you attempt to repair the crack you'll void the warranty. Good luck, Tom