What could be dangerous about a big basin of water? Right?
Scalding / hypothermia
Other than that, you are good to go! Lets start with drowning. The numbers are hard to find, but death by accidental drowning and death by accidental gunshot for children 17 and younger are pretty much on par. In 2011 it was 93 total deaths by drowning in bathtubs with over 75% of the deaths by children age 4 and under. In Japan almost 5,000 people died by drowning in 2014, primarily people over the age of 65. That's a sad story for another article.
So here are the CPSC's guidelines for safely bathing your child:
Never leave young children alone near any water or tub or basin with fluid. Young children can drown in even small amounts of liquid.
Always keep a young child within arm's reach in a bathtub. If you must leave, take the child with you.
Don't leave a baby or young child in a bathtub under the care of another child.
Now, for scalding. There are tragic stories all over the internet of children and babies who were NEARLY burnt to death in scalding bath water. It's sad and sickening. I'm not posting a link and I don't recommend looking. When bathing children and the elderly, the water should not go over around 100 degrees. How do you know what temperature the water is? Try a bathtub water temp thermometer. I love this rubber duckie one, but they come in all sorts of shapes and sizes and run around than $10. Some have audible alarms and some just show the temperature.
The bathtub overflowing is not really a danger to humans, but it's dangerous to your lease agreement. Set a timer to check on the bathtub when filling it. It's easy to forget. Or get an overflow alarm where you set your desired depth and then the alarm goes off when the water is at that depth. Remember, you need a few inches below the top for water displacement. It will vary by your size and the size of your tub, so experiment, don't guess. Plus, if you are one of those who dabble with the temperature of the water, you will need some room to add hot or cold.
Finally, toxicity. A dirty bathtub can be dangerous for those who are susceptible to infections such as those with severe decubitus, burn patients, skin allergies, skin sensitivities, post child birth, proctology issues, etc. For these people an immaculate tub is important and so is the ability to bathe. In this situation, you must deep clean the tub and then MAKE SURE that the chemicals you clean with are completely rinsed out before filling the tub up. Or you could use a TUBSKY™ and know that the tub is clean and chemical free and then throw it in the washer after bathing to clean it for the next bath.
Stay safe and bathe clean!