In most physical cases, the elements of a group can be represented by unitary matrices. Why no time-reversal? In Dresselhaus's group theory page 19, a theorem writes:

Every representation (of a Hamitonian's group) with matrices having non-vanishing 
  determinants can be brought into unitary form by an similarity
  transformation.

On page 21 writes:

On the other hand, not all symmetry operations can be represented by
  a unitary matrix; Time reversal operator is represented by an
  anti-unitary matrix.

My question is, since time reversal operator represented by a matrix, which has the determinant of a non-zero value(?). Is this a contradiction?
 A: I think this passage is not very good (By the way, here is an online version with slightly different paging: http://web.mit.edu/course/6/6.734j/www/group-full02.pdf).
If you have a look at the proof, what they actually does is the following:
Theorem: Every representation of a finite group with matrices having non-vanishing determinants can be brought into unitary form.
I don't see how they can get rid of the finiteness assumption (unless he uses some sort of compactness/boundedness), so they should state this - it's at best sloppy. In particular, the counterexample of katz is then invalid, since the group of integers is clearly infinite.
Anyway, there are well-known theorems along these lines such as:
Theorem: Every representation of a finite group is unitarizable. (see e.g. here)
Theorem: Every finite dimensional representation of a compact group is unitarizable. (follows from e.g. here)
However, when they speak about representations, they always mean linear representations. Which brings me to the time-reversal symmetry:
This is antiunitary and therefore in particular antilinear. It can therefore not be represented by a matrix - since a matrix represents by definition a linear operation. Therefore, the theorems don't apply.
A: The claim on page 19 seems incorrect without further clarification.  For example, the group of integers $n$ acting by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ cannot be made unitary. Perhaps there are further restrictions on the group or on the action? Perhaps compact group, or action by Hermitian matrices?
A: The seemingly contradiction is from the fact that time-reversal operator can't represented by a matrix. It is a matrix times a conjugate operator, therefore talking about its determinant is meaningless.
