How to avoid apathy in the absence of self?
Very good and relevant question. Based on hearsay and my own experience, I suspect this problem is rather common with serious, sincere practitioners.

On one hand, it is totally normal to keep getting dispassionate and detached as you grow your "eye of no-self". After all, the Buddhist realization was described by the Buddha himself as "dispassionment, disillusionment, disenchantment, nibbana" (or something like that).

On the other hand, putting too much faith in this idea of no-self (and its concomitants of "no future to strive for", "no sentient beings to save" etc.) as some kind of Final Truth indicates incomplete understanding of Dharma, at best.

In fact, my teacher strongly insisted that "3D vision occurs when you look with both eyes". Meaning, the anatta view and the "atta" view - are the two eyes. In one, there is no sentient beings and no suffering, no Path, no Attainment etc (quote the rest of the Heart Sutra) - and in the other there are countless sentient beings who suffer and need help. Enlightenment is called "the knowledge of all modes" exactly because it involves going beyond a single perspective, and seeing all perspectives (all "worlds") at once. Enlightenment is the pinnacle of empathy because it involves appreciating all perspectives out there, including perspectives of the most unfortunate, confused, and suffering sentient beings.

As your realization of Emptiness matures, the idea of single correct point of view, or the idea of absolute truth should start making less and less sense to you. At the same time, being able to see things from other perspectives - and empathize their pain, which for them is certainly real! - is the foundation of enlightened compassion.

The dispassion that comes from liberation from any single point of view (atta, anatta, and all the others) and the compassion that comes from a realization that most of the sentient life is locked into a single point of view are the two signs of correct realization.

The above is about the View. As for the Action, this sense of apathy probably comes from an established habit of living a spiritually sterile life. The same exact principle is at work here. While you cling to a single point of view (that of spirituality), you try to be as honest to that as possible and this leads to this sterile lifestyle. Once you realize that Emptiness means no absolute reference point and this realization sinks in to the level of daily life, you will no longer be restricted by one model of behavior. Then your life can start getting back the colors -- without the passions that came from taking it all too seriously.