Based on my experience, here are the most likely culprits and and some ideas.

The biggest culprit are the fragrance/essential oils. It's really hard to get the right amount. Too little and folks will complain. Too much and people will get burned. If you really like a soap and are getting burned, you might try melting it in a double boiler, low heat or a crock pot on low. If hard enough, grate it with a cheese grater. You may need to add a small amount of water. This will usually cause the fragrance oils to evaoporate.

The next culprit is shaving technique. I've had soaps that used to burn me, yet don't do so now. It took me a while to get a handle on 'shave the cream only and the whiskers will take care of themselves'. Proper shave prep figures in here as well.

The next culprits are things you can't control and can't use the soap. First, you may be particularly sensitive to an ingredient, usually a fragrance or essential oil. I have great luck with alcohol based after shaves. I can't use the ones that have witch hazel and aloe added because they burn me.

Finally, The basic action of making soap is combining fat/oil with lye. Every batch needs to be ph tested to ensure that there was enough fat/oil to completely saponify the lye. It gets ugly quickly if this step is missed and there is a problem.