Now, I've seen people pocket hosts before and immediately leave the Church, and I've seen other people who seem to treat as a light snack that they leisurely nibble. I'm more concerned with the former, but the latter also betrays poor formation. I left my pew to speak with the young man in question. As it turned out, he had consumed the host and didn't realize that he wasn't really supposed to join in. I caught up with him later to explain a bit and to invite him back (as well as to apologize for putting him on the spot, although I don't know that anyone else even noticed).
I've since decided that I'm going to embark on a book project in my <fe>voluminous spare time</fe>—specifically, a guide for non-Catholic visitors to the Catholic liturgy. The book would explain some basics about Catholic worship, note some of the distinct doctrines pertaining to statuary, saints, and, of course, explain why non-Catholics or non-Orthodox should not receive the Eucharist. To phrase this question correctly, why do Catholics reserve the Eucharist for Catholics and Orthodox only?
Note that phrasing because it is important. I did not write, "Why do Catholics exclude non-Catholics from communion?" The reason for my choice of phrasing stems from the intent of the Church. It is not meant to exclude but to make clear that communion means something. It is not merely sharing a common ritual or empty symbol but a sharing of an essence that binds us together in faith. That bond is part of what we call "communion," but there are external manifestations in other aspects of Catholic doctrine and liturgy. In addition, there are serious theological differences that make full communion elusive. Eucharist, being the foremost sign of communion in the Church, naturally stands as representative of commonality of doctrine, liturgy, and faith. Where there is no commonality, there is really no communion, despite how we might act ritually.
Protestants are fond of attributing to St. Augustine the following statement when discussing the matter of doctrine: "In essentials unity, in non-essentials liberty, in all things charity." Sadly, there seems to be considerable disagreement about whether he actually made this statement. However, even had he made the statement, one is still left with a question: who decides what is and what is not essential?
Continuity of Apostolic succession (episcopacy) is also a critical matter, as the validity of some sacraments in a Church depend on this continuity, which is why only those churches considered to have valid sacraments may intercommunicate, given the appropriate circumstances.
For Catholics, a proper understanding of authority and of the Eucharist are essential. For the early Church, Apostolic authority and the uniqueness of the Eucharistic mystery were essentials. These cannot be dismissed as minor details for us because we have always considered them essential.* Always. To ask Catholics to dispense with these elements (and others) would be like asking Jews to recast their understanding of themselves as a people without any reference to the Exodus or Passover. They are foundational aspects of the faith. They may not be the only foundational aspects, but they are foundational and essential nonetheless.
So very high on the list of reason for restricting intercommunication is the serious matter that we are not in communion so long as we have such divergent opinions on basic matters of faith. It doesn't mean Catholics hate people of other faiths or wish to see them outside of God's grace. It is a recognition, and also a respect for, the fact that we have serious doctrinal differences that we cannot just brush aside. We take non-Catholics seriously when they say that they don't believe what we believe, particularly concerning the Eucharist.
However, that's not the only reason. There is also a matter of Christian charity. The Church has always maintained that we must prepare ourselves to be worthy** to receive the Eucharist by examining our consciences to ensure that we are not guilty of any grave sin that has not been confessed. As St. Paul states in 1 Corinthians 11:27-29:
Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of profaning the body and blood of the Lord. Let a man examine himself, and so eat of the bread and drink of the cup. For any one who eats and drinks without discerning the body eats and drinks judgment upon himself.
Paul notes physical illnesses and even death among those who have eaten unworthily. At very least, we are concerned about spiritual health. Receiving Eucharist in a state of mortal sin is itself gravely sinful. We as Catholics do no good to someone by encouraging them to an act that may, given the circumstances, be sinful. If we do, then we are also morally liable for whatever spiritual damage ensues.
*I write this in full recognition that some Catholics don't understand this teaching or, if they do understand it, don't accept it in the way that the Church means it. There are also people who understand it in a flawed fashion.
**Of course, we can only be worthy once we have accepted Christ's forgiveness for our sins. So worthiness is contingent on Christ's grace, not on our actions.
***<fe> = irony