We are new homeowners. Our 11-year-old 2nd-floor water heater was leaking and the first plumber who came told us he would need to come back with a partner to take it away. I asked if I could drain it in the meantime and he said that doing so could disturb the sediment at the bottom of the tank causing a much worse leak and likely water damage. He said he usually removes leaking water heaters without draining the water till they're outside the house.
This sounded like a recipe for disaster, and it probably would have been. As commenters and the correct answer below say, if you're ever told this, at the very least:
GET A SECOND OPINION!!!
The plumber who came today said that in 20 years he has never had problems draining a water heater, and he drained ours without incident. It was then easy to remove and replace.