Can you go through a McDonald's drive thru in a taxi?

Sure.

Yes. And the meter is running…

Yes, but it will cost you.

Neither the driver nor the restaurant will have any problem with that.

Your paying for it, sure.
If ya want the cabbie will even wait for you at a movie,but again you will pay for it.

I have several times. I have had some accept as a tip I pay for their order.
Once I had a cabbie ask about my order what is a McDouble. He was an immigrant from Africa, and had no knowledge of ordering them because they sell those cheap, but many don’t even have it on their boards.
I explained it to him, and he was very happy to have this insider info. Lol.
It works best to discuss in advance what you want to order, and whether you are willing to include something for the driver, so when it is time to place the order they know what to say. Otherwise they have to line up the rear window for you.

Yeah, I don’t see why not. You’re a paying customer.