Infrared (IR) excesses around K-type red giants (RGs) have previously been discovered using Infrared Astronomy Satellite (IRAS) data, and past studies have suggested a link between RGs with overabundant Li and IR excesses, implying the ejection of circumstellar shells or disks. We revisit the question of IR excesses around RGs using higher spatial resolution IR data, primarily from the Wide-field Infrared Survey Explorer. Our goal was to elucidate the link between three unusual RG properties: fast rotation, enriched Li, and IR excess. Our sample of RGs includes those with previous IR detections, a sample with well-defined rotation and Li abundance measurements with no previous IR measurements, and a large sample of RGs asserted to be Li-rich in the literature; we have 316 targets thought to be K giants, about 40% of which we take to be Li-rich. In 24 cases with previous detections of IR excess at low spatial resolution, we believe that source confusion is playing a role, in that either (a) the source that is bright in the optical is not responsible for the IR flux, or (b) there is more than one source responsible for the IR flux as measured in IRAS. We looked for IR excesses in the remaining sources, identifying 28 that have significant IR excesses by ~20{mu}m (with possible excesses for 2 additional sources). There appears to be an intriguing correlation in that the largest IR excesses are all in Li-rich K giants, though very few Li-rich K giants have IR excesses (large or small). These largest IR excesses also tend to be found in the fastest rotators. There is no correlation of IR excess with the carbon isotopic ratio, ^12^C/^13^C. IR excesses by 20{mu}m, though relatively rare, are at least twice as common among our sample of Li-rich K giants. If dust shell production is a common by-product of Li enrichment mechanisms, these observations suggest that the IR excess stage is very short-lived, which is supported by theoretical calculations. Conversely, the Li-enrichment mechanism may only occasionally produce dust, and an additional parameter (e.g., rotation) may control whether or not a shell is ejected.