How do you handle feedback and criticism from the Product Hunt community?

Thank them. It's their view. Filter them. See if they brought up specifics, if they are your target audience, etc. Act on them. Break down feature requests into use cases, ask further questions, connect to demo your product, implement quick fixes, add product ideas to your backlogs if they warrant that.

If somebody is taking their time out for free to help me improve, I am more than grateful. And then I try to reflect on it and learn from it. More often than not, there will be an insight I have overlooked. Having said that, i do filter out some feedback which doesn't have a clear logic to it.

Just smile and respect the person and evolve. :)