At some churches, non-members are not allowed to take communion and this can seem insulting or even sacrilegious. Why would any church not allow a believer to take communion? It depends on what that tradition thinks communion is all about.
The answer is easier to understand once one realizes the difference in significance that traditional churches place on communion vs. the Zwinglian/Baptistic memorial-only view. For most traditional churches, communion is not just a symbolic-memorial act shared by any given group of Christians. rather, it is equivalent (in a mysterious way) to the very Body of Christ. Thus, as church members (the Body of Christ) partake of communion (the Body of Christ) they become the Body of Christ!
This view of communion is that it is not only the “high point” of the traditional service, it is the point at which Christ truly becomes present, and John 6 (not just Matthew 26) is to be fulfilled and the Body of Christ made one. Because of this, it would be a serious offence to take communion in a church when one is not really part of the commun-ity (especially if one is under discipline of ex-commun-ication). It could be seen as lying to act as though one is a part of a church that they are not.
It could also be dangerous – Paul, for example, says they could die (1 Cor. 11:30). So for a church to let just anyone take communion is to put their lives at risk. (In fact there are times when even members are not to partake.)
Because the Bishop / Priest / Pastor is in charge of dispensing the Eucharist meal, it is his job to insure that communicants are legitimate. Non-members could not receive this oversight, because he would not know them. And, even though communication between churches is not always black and white (for example, there is some orthodox-catholic overlap sometimes), for a Christian who rejects communion with a given body, it would be disingenuous at best to take part in the very act that unites them and communicates their unity.
Because this notion of communion is not prevalent among more modern baptistic churches, it can be hard to see – but the above ideas are part of the communion “package” for traditional churches. (Consider how a Baptist would feel if a non-believer got baptized just because that’s what people do in that church. The unbeliever may not mean anything bad by it, but it would not be appropriate.)
The safest bet is to just ask beforehand. And if one is not allowed to partake in communion, one should not be insulted – the church is just trying to protect people from danger and sin.