How not to solve P=NP?
I'd say the most well known barriers to solving $P=NP$ are 


Relativization (as mentioned by Ran G.)
Natural Proofs - under certain cryptographic assumptions, Rudich and Razborov proved that we cannot prove $P\neq NP$ using a class of proofs called natural proofs.
Algebrization - by Scott Aaronson and Avi Wigderson. They prove that proofs that algebrize cannot separate $P$ and $NP$


Another one I'm familiar with is the result that no LP formulation can solve TSP (It was proved by Yannakakis for symmetric LPs and very recently extended to general LPs). Here is a blog post discussing the result.