How do we respond to customers who call us for support we don't offer?

If people are only occasionally getting through, I'd err on the side of generosity. Say, "I'm afraid we don't offer support on {xyz} which is a free product, but let me see if I can help you out..." It's always valuable to engage with users, so while it's manageable look at it as an investment in marketing.

But shouldn't you be thinking about the signal you're getting? The fact that the free product is encouraging people to reach out to you even though that means some detective work on their part must tell you something. Maybe it's telling you that you need to divest the product. Or maybe it's telling you that there's an opportunity to build on it. Or maybe it's telling you that in this case you may have misjudged the balance between cost and value. I'd want to spend time figuring this out before I made any changes to customer support policy and practice.

Why not setup some resources to use this opportunity to convince your free users to upgrade?

Set up a phone call with your free customer at a time convenient to you. Help them out and tell them that you're helping them because they ended up calling you (so why not), and brief them that this is the kind of help they would actually benefit from if they upgraded to the package with support.

Though, this strategy depends on the time and expertise required to provide support. Depends on the nature of business. I wouldn't let a customer calling me go unconvinced. I would help them for as long as I can. For something very specifically technical, I would ask them to upgrade beforehand.

Since it is a recurrent question, you should have a already prepared response. I would just choose one of the following two:

a) Thank you very much for your call. I really appreciate the effort you have made to reach us. Unfortunately free-product-name does not have any kind of direct support and we have decided it will not have it in a foreseable future. [Now that the information is given, you can chat or discuss any other matter with the customer]

b) Thank you for calling. Right now free-product-name does not have the kind of direct support you are asking for, but we are making an in-deep analysis on whether it would be profitable to give the kind of servide you are asking for. [From here on you can ask questions to help you in the analysis OR you can tell the amount of money that such a service would be offered for].

IMHO the most important part of answering this kind of calls is to do it without hesitation and using the least possible amount of time and energy. Make a decision beforehand (essay your speech if you have to) and tell the customer straightforward which is the situation.

Assuming you're a business, listen to what your users are asking for, and figure out if there's a viable way of providing it. Perhaps those customers would be willing to pay for a supported version of the product, or just the support itself.

If you choose not to do this, be ridiculously transparent about it and let the community provide its own support.

Just respond honestly:

"I'm sorry but that is a free product and we do not offer any support for it. Please refer to the online help file, or upgrade to our paid product XXX."