Is it important to feel the chest while doing bench press?
Since the primary purpose of the bench press is to develop the chest you obviously want to be recruiting those muscles as much as possible during the lift.
However it is worth noting that the pecs are a relatively large muscle group that is unlikely to be your first point of failure during a lift. Your triceps and anterior deltoids are also involved in the bench press and are more likely to fail first.
The solution is to lift light weights and focus on contracting your chest, as opposed to loading heavy weight and just pressing as hard as you can.
Over time, you will learn to contract your chest more during the bench and your pecs will receive more benefit than they might have during the beginning of your training. Anecdotally, when i first started training my chest didn't really get sore from benching, but over time I've learned to use the desired muscles more.
Part of how I know this is because I recently spent three weeks in Boston and only had access to 50 pound dumbbells. When I got back and went back to pressing with a barbell my chest was sore the next day because the neuromuscular connection has been built from years of lifting.
So...yes and no. Don't worry about it too much but make sure you are really focused on squeezing your chest during all of your lifts if that's what you want to develop