Is it correct to write a phrase like "The guy that is responsible is David"?
There isn't any rule analogous to the "double negative" rule for be verbs. If you're worried that you're producing two verbs for the sentence, let's take a look at what you're saying:
The guy that is responsible is David.
Here, that is used to mark a non-essential clause. These phrases provide extra information, and can be removed from a sentence without compromising the grammatical structure of the sentence. Let's do that.
The guy is David.
A rather simple sentence, and perhaps a bit awkward, but technically correct, nevertheless. This shows that your non-essential clause wasn't making your sentence incorrect. In proper context, it could even be usable:
"We need a guy who really knows his way around Oracle systems." "'That guy' is David."
Here, the first speaker desires an employee who's good at working with a specific piece of software. The second employee points out that they already have someone hired with the proper qualifications. I changed "the" to "that" because it is more idiomatically appropriate, and most likely how you'll see a sentence with that structure used.