It might be that at the point in the book where you encountered this rule the theory is not yet far enough developed to cover this phenomenon -- if you have not yet heard of stuff like movement and gaps and traces, it might be best to just be patient and wait until it is introduced later. If by the Carnie book you mean the one addressed in my comment, from what I can tell it is not dealt with in the book at all; in the third edition of the book, it will be covered in ch. 12.2 "Relative clauses" -- though interestingly, that chapter only cites examples of relative clauses where it's the object that is turned into a relative pronoun and not, like in your example, the subject, but the general principle works the same for both NP positions. So let me adapt the idea to the theory as presented in the 2006 version of the book:
The "that" in the relative clause "that lies in the table" is not is not the same "that" you have in e.g. "I believe that the book lies on the table". In the context of "the book that lies on the table", "that" is not a complementizer which serves to introduce a subclause, but a relative pronoun that stands for an NP. This is maybe more convincing once you realize that "that" can be replaced by a wh-pronoun "which" or "who": "the book which lies on the table" is semantically and syntactically the same, and it's clear(er) that the wh-pronoun "which" refers to and acts like an NP.
Here, that NP is "the book": "that" takes the place of the subject "the book" that complements the VP "lies on the table". Hence, it is assumed that "that" starts out as an NP in SpecT position, and then is moved to into SpecC, leaving a gap with a trace (t) at its original position:
[NP the book [CP [NP that_i] [TP [NP t_i] [VP lies on the table]]]
Under this analysis, "that" serves both as a subject NP to make "lies on the table" a TP and as a specifier to the CP, while the C head is empty.
It works the same if the relative clause is about the object, that is, instead of the subject, "that" takes the role and starting position of the object of the embedded sentence, where the subject is some different NP that remains present in its original position, and instead the object is moved out of the complement of the verb to, again, the specifier of C:
[NP the book [CP [NP that_i] [TP [NP you] [VP [V read] [NP t_i]]]
Again, the relative pronoun "that" is an NP which starts out in the position of the object so we get an intact TP, and is then, in order to form a relative clause, moved out to the SpecC position, while the C head is again empty.
Relative clauses where the relative pronoun is about the object also make it more obvious why the movement is at all necessary: In the first sentence, we would get just the same word order if we assumed that "that" just remained in subject position, so we might wonder why we want to make the complicated assumption of moving it away to somewhere different that leaves us with the same word order as before. But in object relative clauses, we do have to move the "that" away from its sentence-final position in order to get it in front of the subject "you", and it makes sense to treat the two very similar cases of object vs. subject relative clauses uniformly and say for the sake of generalizability that relative pronouns are always moved into SpeC, no matter if they refer to the object or the subject of the sentence -- which gets us exactly the structure outlined in the first gray box.