When I first starting studying differential geometry, I asked my lecturer a question about smooth manifolds that didn't admit a partition of unity. He promptly told not to worry about such objects as they were only studied by the extremely eccentric. I would like to know if this is true, ie, does anyone study manifolds that don't admit a partition of unity (not whether such people are eccentric).

The answer to your stated question ("Does anyone study nonparacompact manifolds?") is certainly yes. Here are a few papers which do just this:
Among other things, Gauld references that there are two paracompact and two nonparacompact 1manifolds, and $\aleph_0$ paracompact and $2^{\aleph_1}$ nonparacompact 2manifolds. (That's a lot!)
Milnor entertains nonparacompact manifolds. In particular he constructs a (necessarily nonparacompact) surface with uncountable fundamental group. Milnor also says: "The main object of this exercise is to imbue the reader with suitable respect for nonparacompact manifolds."
The existence of perfectly normal, nonmetrizable (hence nonparacompact) manifolds is shown to depend upon one's settheoretic assumptions. And so forth. I could find 10 more papers without much effort. I'm not sure I could find 100. (A MathSciNet search with "manifold" and ("nonmetrizable" or "nonparacompact") in the anywhere fields doesn't return many hits.) So some serious mathematicians take nonparacompact manifolds seriously enough to write some papers about them. On the other hand, although one could use more complimentary language than "extremely eccentric", your lecturer's take on nonparacompact manifolds seems to be an accurate reflection of how most geometric topologists feel: they seem mostly to be used as a source of counterexamples and to be of interest to general and settheoretic topologists. 


In my experience the answer is not really. My favorite example of a Hausdorff manifold which is not paracompact (which incidentally is also my alltime favorite counter example to most pointset topology questions) is the Long Line. Since it is not paracompact, it does not admit partitions of unity. All of it's homotopy groups vanish, yet it is not contractible. If I remember correctly, its tangent bundle is nontrivial. Even if people don't study these, then it is still important to know about the type of things which don't satisfy the usual hypotheses. You never know when you might run into one or need such a counter example. 

