Do lobsters feel pain?
It is not fully known. The lobster nervous system is decentralized, having a brain and a series of ganglia, and has even less neurons than even that of an ant. However, they also possess endogenous opioids.
In 2009 and 2013, Elwood et al had demonstrated that crabs (who are considered Decapods, along with lobsters and shrimp) can learn to associate electric shocks with an unrelated cue. In the 2009 study, hermit crabs that were shocked when entering one shell often forfeited the shell. In the 2013 trial, crabs were shocked when entering a dark shelter (in which they preferred), and many of the crabs escaped and did not return after being shocked. This demonstrates that they are able to learn to avoid electric shocks. However, trials on lobsters have not been conducted.
All being said, this is still not 100% proof that they feel pain, at least like we do. It is still better to give them the benefit of the doubt and destroy their nervous system quickly, or to refrain from eating them on the next meal.