While discoursing on who can receive the Eucharist sacramentally (Summa Theologiae III, q. 80, a. 3), St. Thomas Aquinas describes three cases where the one consuming the Eucharist does not receive the Eucharist sacramentally: when the Eucharist is consumed by an unbeliever, an animal, or by one who does not know it to be the Eucharist, for instance, if he thinks that the host is not consecrated. About an unbeliever who receives the Eucharist:
Even if an unbeliever receives the sacramental species, he receives the body of Christ under the sacramental sign. Therefore, he eats Christ sacramentally, if "sacramentally" refers to that which is eaten. But if it refers to the person eating, then properly speaking he does not eat sacramentally, because he does not use what he receives as a sacrament, but as simple food. Unless, perhaps, the unbeliever intends to receive that which the Church bestows, although he does not have true faith regarding the other articles or even regarding this sacrament (Summa Theologiae III, q. 80, a. 3, ad 2).
About an animal that consumes the Eucharist, or a person who does so unknowingly:
Even if a mouse or dog eats a consecrated host, the substance of Christ's body does not cease to be present under the species as long as these species remain, that is, as long as the substance of bread would remain [if it were bread], just as happens if it is cast into the mud… and yet it is not be said that the brute animal consumes the Christ's Body sacramentally, because it is not unable to treat it as a sacrament. Consequently, it does not consume the Body of Christ sacramentally, but only accidentally, just as he who consumes a consecrated host not knowing that it is consecrated (ibid., ad 3).
If I rightly understand Aquinas's position, it could be described in the following manner: the Body of Christ is present under the species of bread as long as the species of bread remains, and in this enters into any person or animal who consumes this species; but, if they do so without at least an implicit intention to receive the Body of Christ, they are not united sacramentally with him; the Body of Christ is present in them in the same way that the Body of Christ is present in a ciborium, or perhaps in a vessel of water that is dissolving the species of bread (somewhat analagous to the process of digestion dissolving the species).
In a similar manner, one who knows that he is consuming the Eucharist, but does not intend to do so sacramentally, does not in fact receive sacramentally. There are two quite different ways this could occur: (1) someone could ritually receive Communion, e.g., during a Mass, while intending not to receive sacramentally, to be sacramentally united to Christ; this would be a grave sin against the Sacrament; (2) someone could consume the consecrated species in another context; e.g., a minister of the Eucharist consuming a host that falls on the floor, someone consuming remaining hosts or the Blood after Mass, or purifying the sacred vessels. This seems to me not to be a sacramental reception. One liturgical argument that could be advanced for this is that an institute acolyte may take and carry the chalice to the credence table (without receiving it from the priest) and purify it there. If such purification is to be considered a sacramental reception of communion, then this would be self-communication, which is generally forbidden.
Though the distinction between "receiving that which is the Sacrament" and "sacramentally receiving the Sacrament" may in this case seem a subtle one, I wonder whether it might not be a valuable point of contact with some of the protestant understandings of the Eucharist and their emphasis on the role of faith. Admittedly we cannot overlook the differences; the Council of Trent expressly rejected Luther's position that the Real Presence is only there in the consumption of the Eucharist (in usu) (Session 13, Canon 4); but such points of contact, even if they do not involve simple agreement, are still important.