We are going to add a separate tub to a bathroom with an existing shower. The plumber plans to connect the tub drain in series to the existing shower drain. The tub would have a P-trap adjacent to its drain and then feed into a vertically oriented sanitary tee. The bottom of this sanitary tee would continue downward into a P-trap for the shower, and the top of the sanitary tee would continue upwards to the shower drain. The shower P-trap connects into an existing vented waste line about two feet away.

I don't have any plumbing experience, but after some reading I have at least three concerns with this plan: (1) the tub would seem to be double trapped, (2) the shower drain would seem to be a vent for the tub drain, and (3) when we drain the tub it seems like it could fill the shower temporarily. I think there needs to be two separate lines connecting to the existing vented waste line. The tub drain would be within 2 1/2 feet of the shower drain and 5 feet of the waste vent line.

A sketch is attached.

Your drawing failed to make it.
The plumber plans to connect the tub drain in series to the existing shower drain. You can't do that! Double trapping is outlawed in all plumbing codes.
It makes no never mind how he plans to drain and vent the fixtures. The double trap will end all that. You don't have a "plumber" you have a handi-man that doesn't know code. Get someone in there that knows what he's doing. Good luck, Tom

I agree with Tom. Only Double Trapped fixture can be sink.

Only Double Trapped fixture can be sink.
Let's clear that up. While a kitchen sink can have two traps, one serving each tub you can not install two traps in series thus double trapping the sink. I'm sure this is what Milo meant, regards, Tom

Thanks for making it clear, Tom