There are several churches which use the idea of transubstantiation; but the Eastern churches (Eastern Orthodox, Oriental Orthodox, and Assyrian Church of the East) share a similar view on how it works. I'll therefore split my discussion into two parts, the "Eastern" and "Western" views.
The Eastern churches have occasionally used the term "transubstantiation"; sometimes (being Greek-speaking in origin) they would use the Greek-derived equivalent "metousiosis". They don't get very specific about how transubstantiation works technically. The Longer Orthodox Catechism (articles 339 - 340) states:
The bread and wine are changed, or transubstantiated, into the very Body of Christ, and into the very Blood of Christ. ...
In the exposition of the faith by the Eastern Patriarchs, it is said that the word transubstantiation is not to be taken to define the manner in which the bread and wine are changed into the Body and Blood of the Lord; for this none can understand but God; but only thus much is signified, that the bread truly, really, and substantially becomes the very true Body of the Lord, and the wine the very Blood of the Lord.
In other words, they're just saying they believe that the bread and wine "become" the Body and Blood of Christ, but they don't know (and really don't care) how; it is a mystery. The Eastern churches in general don't do as much philosophical speculation in their theology as the Western ones historically have; they have an entirely different approach. Certainly, they are aware and believe that the bread doesn't become body parts in some sort of literal sense. Thus, they're not concerned with the possibility that the consecrated Hosts become parts of Christ's body and add to his weight.
The Western View
The most systematic approach we have to an answer here is probably through the work of Thomas Aquinas, particularly his treatise Summa Theologica ("A Summary of Theology"). Unfortunately, it's very difficult to understand what Aquinas says about it simply by reading the book, without knowing a good bit of medieval philosophy, particularly medieval interpretation of Aristotelian philosophy. Aquinas' take on whether Christ's weight (or other aspects of his body) is really present in the consecrated host is this:
As stated above, any part of Christ is in this sacrament in two ways: in one way, by the power of the sacrament; in another, from real concomitance. By the power of the sacrament the dimensive quantity of Christ's body is not in this sacrament; for, by the power of the sacrament that is present in this sacrament, whereat the conversion is terminated. But the conversion which takes place in this sacrament is terminated directly at the substance of Christ's body, and not at its dimensions; which is evident from the fact that the dimensive quantity of the bread remains after the consecration, while only the substance of the bread passes away.
Nevertheless, since the substance of Christ's body is not really deprived of its dimensive quantity and its other accidents, hence it comes that by reason of real concomitance the whole dimensive quantity of Christ's body and all its other accidents are in this sacrament.
(Third Part, Question 76, Article 4)
This, like a very dry Host, is rather hard to swallow all at once. Essentially what it means is this: The weight of a physical entity is one of its accidents, one of the things that just happens to be true of it in particular, but doesn't change what kind of thing it is. (So my weight could change without changing whether I'm human or not.) Since transubstantiation changes the substance of the bread and wine, but not their accidents, transubstantiation does not directly give the bread and wine the weight, or height, or eye color, or what-have-you, of Jesus.
However, there's a surprise waiting: the doctrine of concomitance. This simply says that nothing about Christ, including body, blood, and what-have-you, is really separable from anything else about Him, so that if He's present in some sense, he's present in all senses. Thus, through the "real concomitance" approach, his weight is there. The weight of the Host, however, is a different accident—it's "attached to" the other accidents of the bread, which remain the same, and so it doesn't alter the weight of Christ.
Clear as mud? I thought so. Let me know what I can do to clarify it.