When referring to phrases such as "that book", would it be considered a DP or a NP? I think it should be considered as a DP but I am not sure how to prove it using our given data. Some data we're given are ellipses ("I like Max's house and Max likes Jimmy's), NPIs ("No student will miss any class vs *Some student will miss any class") and required determiners ("They read a book vs *They read book). I am not entirely sure how to use this given data to argue for/against the hypotheses. Anything helps!
First, a note: this isn't the only possible way to answer the question. You can also argue for it being an NP with special restrictions that mean it can only combine with the null determiner. There are also some theories which don't use DPs at all, just different flavors of NPs with different restrictions. But to me, that adds extra complexity and headaches for no real gain; I think the DP analysis is much simpler.
I would argue that it's a DP. I don't have all of your data, but here are a few ways I would show this using native speaker intuition.
Coordination can happen at two separate levels, "inside" or "outside" the demonstrative. This indicates that there's another phrasal level inside "that book".
I love these [dogs and cats].
I love [these dogs] and [those cats].
These phrases can coordinate with proper nouns, which are usually considered to be DPs. Bare nouns can't.
I've been feeding [Nico] and [this other cat].
*I've been feeding this [Nico and other cat].
They can appear in places where bare NPs aren't accepted.
I pet [the cat].
*I pet [cat].
I pet [this cat].
They can't be combined with other determiners.
Look at the [cat].
*Look at the [this cat].
All in all, it seems to act much more like a DP than an NP.
I like that book.
In the DP theory, the determinative "that" is head and the noun is the dependent. The demonstrative determinative "that" is just as much a determiner here as "the" is, so there is no structural difference between the book and that book; they are both NPs.
There are a couple of facts to support the NP analysis. One is that it is the noun (or nominal) which defines the selectional properties of the phrase. For example, a verb like assassinate selects a human head, whereas there is no verb in English which selects an object phrase determined by the, as opposed to no. This is because the basic semantic function of the determiner is to indicate whether the phrase is definitive or indefinitive, and this is independent of the role the phrase otherwise plays in the larger construction in which it occurs.
The second reason for taking the noun as head is that while there is a wide range of ordinary NPs that contain no determiner (NPs like Kim, money, women, they), NPs that do not have a noun as ultimate head (NPs like both, several, the largest) are highly restricted in their form and use.
The conclusion is that the determiner is a kind of dependent – contrasting with complements and modifiers. And, as noted above, it need not be realised by a determinative, but can itself have the form of an embedded NP, in either genitive or plain case.
Ref: CGEL, Huddleston & Pullum.