Q:

What does 'hermeneutic analysis' mean for non-literary material? 

I am reading Exploring Robotic Minds by Prof. Jun Tani in which he describes Heidegger as having performed a hermeneutic analysis of the problem of the subjective vs the objective. 

Heidegger just could not accept the unconditional prior existence of the cogito.  Nor could he accept an ideal and logical representation of the world that the cogito supposedly constitutes.  Instead, he raised the more fundamental question of asking what it means to be human, while avoiding tackling directly the problems of cogito versus perception, subjectivity versus objectivity, and mental versus material.  It is important to note that Heidegger sought not to obtain an objective understanding of the problem but rather to undertake a hermeneutic analysis of it. 

So my question is: 

what does hermeneutics mean when you aren't analysing a thing (text or other work)
what is meant in the specific case I mention? 

A:

The easiest account of this I found is the following by Gadamer, as (re)stated in Philosophical Hermeneutics, p.9:

Heidegger worked out this primacy in his doctrine of the productivity of the hermeneutical circle.  I have given the following formulation to this insight: It is not so much our judgments as it is our prejudices that constitute our being. 

Hermeneutics in classical understanding mean:

To exclude by controlled, methodical con­ sideration whatever is alien and leads to misunderstanding - misunderstanding suggested to us by distance in time, change in linguistic usages, or in the meanings of words and modes of thinking (ibid, p.7)

To make something more out of it: Whereas originally, hermeneutics were about understanding texts (Schleiermacher, Dilthey), Heidegger extended the term on the overall strive and struggle of humans trying to find meaning in what they are confronted with.  Thereby, we cannot give up everything that helped us before when confronted with something new, meaning that we have a limited framework from which we can understand new things, i.e. we rely upon our prejudices (in a literal sense!) -  the hermeneutical circle that constitutes the limits of our understanding. 
This clearly does just as well apply to the rest of our life, not only to artifacts. 
Gadamer, building upon Heidegger's conception of hermeneutics, writes:

The problem is really universal.  The hermeneutical ques­tion, as I have characterized it, is not restricted to the areas from which I began my own investigations.  My only concern there was to secure a theoretical basis that would enable us to deal with the basic factor of contemporary culture, namely, science and its industrial, technological utili­zation. ( ibid, 10-11)

And how this more universal understanding can be reconciled with a classical understanding of hermeneutics is explained as well:

This discussion shows how the claim to universality that is appropriate to the hermeneutical dimension is to be under­stood.  Understanding is language-bound. ( ibid, 15)

This means: Hermeneutics is a mode of methodically excluding misunderstandings, revealing objective meaning (as far as it is possible, see the circle) as different from subjective constraints. 
Conclusion
As understanding of whatever object is only possible through language (beware: not everyone agrees, as it excludes intuitive understanding!),  the differences between written text, art, science, or sensual impressions cannot stop hermeneutics from being applied.  Hermeneutics, in this understanding, is a basic mode of being, mediating between the (objective) immediate and the (subjective) understanding. 
This, the quote states, Heidegger methodically applied to the connection between subjectivity and objectivity itself.  And as the hermeneutical circle prevents us from ever achieving objectivity, Heidegger never claimed to have reached it.