I think you'll be surprised at the results. At the very least your hacker brain will be more informed, especially if you put a dollar figure on your customer support time. Doesn't have to be anything fancy, it could even be minimum wage.
You have probably enough word of mouth now and establishment to build off of with your product that you can get some new customers without it. You can always do trials or free POC periods.
I agree free stuff is neat. I think the other side of the hacker ethos is "figure it out yourself", so I would never expect someone who hacked something together and shared it for free to then provide support for said thing. That's cool if you want to do it, but seems like asking for burnout unless you really like the support side or perhaps are looking at turning it from a free thing into a business, as you seem to be.
However, there's another breed of hackers that enjoys making things more than they enjoy making money from things they make (which involves a different set of skills and interests). In which case, it's also valid to just say, "Sorry it's hard, but I can't commit to formal support."
But perhaps the real tradeoff you're making, is to do more support work in order to allow the free tier?