Non-termination of types in Martin-Löf's Type:Type?
Short answer: yes.

Long answer: For $\mathrm{Type}:\mathrm{Type}$, non-termination at the type level is trivial. You can take a constant $X:\mathrm{False}\rightarrow \mathrm{Type}$. Then if you take the inconsistent term $\bot : \mathrm{False}$ you have
$$ X\ \bot : \mathrm{Type}$$
Which is non-terminating at the type level. You might complain that this has a head normal form, which isn't what usually leads to inconsistency in type theory. In this case you can remember that 
$$ \mathrm{False}\equiv \forall X.\ X$$
and so
$$ \bot\ \mathrm{Type}:\mathrm{Type}$$

Which has no head-normal form. In general, in this system the types and terms are so intertwined that the non-termination always seeps at the type level.

However, there is another system, called $U^-$, also described by Girard in his thesis, which was discovered to be inconsistent by Coquand (A New Paradox in Type Theory). This system is terminating at the type level, as it only has $\mathrm{system}\ F$ types at the kind level, and we know that terms are normalizing in that system (also a result of Girard!).

This means that non-termination at the type level is not necessary for having an inconsistent pure type system (a fact that I found out somewhat painfully after having proven an open question while depending on this fact).