How to avoid apathy in the absence of self? I have noticed in my meditation / mindfulness practice the following: as the idea of self dissolves, the ideas of compassion, happiness, sorrow, and basically everything that a "person" might experience also dissolve. 

This is disconcerting for a few reasons:


I stop caring about the suffering of others. After all, "no one" is really suffering, so there is no "person" to receive compassion. The "person" that I direct compassion to right now, is simply a volatile "soup of phenomenon" that won't even exist a few moments later.
I lose motivation to do things because I don't feel fear the way I used to. Because of my upbringing, fear of consequences was the primary motivation for doing anything. Example: in relationships, fear of letting people down would motivate me to pursue resolution to problems, resolve conflicts, etc. 
I don't enjoy things in life the way I used to. As negative emotions have dissipated, so have positive ones. I no longer enjoy eating a meal with good friends, or a sunset or a dozen other things I used to. No moment is particularly better or worse than any other moment. When I live life one moment at a time, there is nothing to "look forward to" because I am not doing anything I dislike that I am trying to "get done as quickly as possible."


Bottom line: the more I meditate and live moment by moment, the more dull and uninteresting the world becomes. Consequently, I don't know how to get from here to having compassion or even interest towards other living beings. 

Can anyone offer some guidance?
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.