What could cause burning and tingling in the feet?

Most likely two. Things, either neuropathy (inflammations of the nerves) or you may have neuroma in the foot.
A few things. A systemic disease such as diabetes, a back problem like a herniated or bulging disc. A side effect of a medication or radiation treatment given for cancers, alcoholism, or nutrional disorders. Localized problems in the feet such as tarsal tunnel or a neuroma. Both of these are entrapments or pressure exerted on specific nervees.