Is there an affordable experiment which shows chips can't get much smaller?
I think you are right that the Heisenberg uncertainty principle is involved in the limit.  I don't think we are anywhere near the Heisenberg limit.  The limit we are currently approaching is that we currently compute with devices based on atoms.  An atom is about 1 angstrom = .1 nm wide.  We are currently making transistors that are about 22 nm wide, so we're getting close to needing to make 1 atom transistors.  If we want to scale beyond that we're going to need to find ways of computing that involve storing and manipulating multiple states per atom.

The wikipedia article on fundamental limits of computation is a good starting point.  I think what you really want is the Bekenstein bound.

The researchers that I know of in this area are Rolf Landauer, Charles H Bennett, and Seth Lloyd.  (I think Charles H Bennett considers himself a computer scientist while the other two consider themselves physicists.)  Richard Feynman was also very interested in this question and has a book called Feynman Lectures on Computation (which you can read about at http://quantum.quniverse.sk/buzek/zaujimave/p257_s.pdf).  Also @PeterShor is an active member here and is the expert on quantum computation.

Here's a Scientific American article by Bennett and Landauer:

http://web.eecs.umich.edu/~taustin/EECS598-HIC/public/Physical-Limits.pdf

Here's Seth Lloyd's article in Nature: http://arxiv.org/abs/quant-ph/9908043

Here's a web page at Cambridge: http://www.sp.phy.cam.ac.uk/~SiGe/Fundamental%20Limits%20of%20Computation%20-%20Landauer%20and%20Heisenburg.html