In-approximability results in severely restricted graph classes Longest path problem is not polynomial-time approximable to any constant factor in cubic Hamiltonian graphs (Longest path $\notin APX$ unless $P=NP$). I don't know if it remains in-approximable in cubic bipartite Hamiltonian graphs. Maximum independent set is not in $APX$ unless $P=NP$ but it is in $APX$ for cubic graphs. David Eppstein pointed out that it is $NP$-complete to find maximum clique in claw-free graphs and it is not clear to me if it is any easier to approximate than in general graphs. 


I'm interested in optimization problems that remains in-approximable in severely restricted classes of graphs (for instance, cubic planar bipartite graphs or trees of bounded degree).


A problem $L$ is In-approximable means that  $L \notin APX$. It is $NP$-hard to approximate to any constant factor in polynomial-time.
The labelled perfect matching problem, which consists in finding a perfect matching in an edge-coloured graph that uses the minimum number of colours, is not in APX for subcubic bipartite graphs (but it is $2$-approximable if the maximum degree is $2$ instead of $3$). See Monnot for details.