Hot Air Gun for Reflowing a Board?
Could this possibly damage the board? 


Of course.


  What about all the parts? 


Yes. It could certainly damage them too, and most probably would.


  Are there any other pitfalls that I might encounter or would this method work well?


The problem with this method is that it is inherently poorly controlled.  It is potentially able to work and an exceptionally skilled and experienced and well trained operator may be able to achieve somewhat acceptable results some of the time. But most of us would just end up with a work of art or a pile of smoldering slag.

Probablility ~= 1.:  Reflow soldering is an exercise in controlled death. Components and board are heated up hot enough and long enough that they are well on the way to destruction. Manufacturers design parts to meet the stresses of this process with an acceptable margin of safety. If you read up on the reflow process in detail, as you MUST have already done to make this question more than idle time wasting, you will have found that temperature profiles - rates of temperature change, holding times and cooling times and temperatures are all tightly specified. If you can manage the sort of control that this implies over the surface of a PCB containing 250 or so components including fine pitch TQFPs then you are wasting your time in your present role and probably want to enroll as a micro-surgeon or Formula One driver or similar :-). ie it's far too demanding a task for this to have any certainty of working.

Probability ~= 0:  Not everyone is Wouter - he is an extremely experienced and capable engineer. All that said, it is "just possible" [tm] that a consistent approach, well aligned jig, temperature controlled air source etc may be able to do the job quite well. Finding out could be expensive. Or not. Given the very great success achieved by the toaster-oven-PCB-assembly community and the large amount of on-web information available on this method and the relatively low cost of doing it,  I'd expect your TQFP's to thank you profusely for taking that route.



Related:

Spark Fun show you how to do Toaster Oven PCBing - lots of details - MANY photos

Some amateur results 

Lots and lots and lots of PCB-toaster-oven ideas

Open Hardware PCB toaster over project

And more ...

Even a small BGA - an instructable