Scripturally there are only a couple of reasons why someone would not be welcome to the Lord’s Table. One is if they are not a believer. As far as I can tell, and I am open to correction here, the Lord’s Supper was exclusively something observed by the gathered church. It was not something that unbelievers participated in or at least that is not obvious from the text. Again, I might be wrong on that. The other reason to deny someone the fellowship of the supper is a person in gross and unrepentant sin that had not been properly disciplined and restored to fellowship. This comes from the well known passage in 1 Corinthians 5. The context is a man who “has his father's wife” which indicates a man having sexual relations with his step-mother (see Leviticus 18:8) which is pretty gross but Paul expands on this specific situation in Corinth to make a more global point (emphasis added):
I wrote to you in my letter not to associate with sexually immoral people—not at all meaning the sexually immoral of this world, or the greedy and swindlers, or idolaters, since then you would need to go out of the world. But now I am writing to you not to associate with anyone who bears the name of brother if he is guilty of sexual immorality or greed, or is an idolater, reviler, drunkard, or swindler—not even to eat with such a one. For what have I to do with judging outsiders? Is it not those inside the church whom you are to judge? God judges those outside. “Purge the evil person from among you.” (1 Corinthians 5:9-13)Paul states pretty unequivocally here that those who are Christians that are guilty of a variety of sinful behaviors should be banned from sharing meals, which I believe is how the early church understood and observed the Lord’s Supper. This denial of the breaking of bread seems to be the most severe form of church discipline.
So what does that say about the “non-member” in a local gathering? From what I can see, you either welcome someone to the breaking of bread or you refuse them because a) the aren’t a believer or b) they are engaged in open and unrepentant sin and are treated as an unbeliever. Since formal church membership in any form is absent from the Scripture, how exactly does that suffice as a reason to deny someone the Supper? Or disagreement on secondary doctrines? If I hold to the Biblical practice of baptism, i.e. baptizing only professing believers, should I withhold the fellowship of the Table from those “baptized” as infants? In my understanding of Scripture they are tacitly refusing to submit to baptism upon profession of faith. There was a time when I would have refused the Lord’s Supper, at least the ritualized form it takes in the traditional church, from someone “baptized” as an infant with a clear conscience but now? Unless I believe that they are not a believer or that they are openly sinning I don’t see where that warrant comes from. The idea that I would enforce a completely artificial standard (being a "member" of that particular local church) as the entry point for breaking bread is divisive and to be blunt sinful.
The Lord’s Supper is the great unifying symbol of the shared faith of every Christian and yet it is most often something that is either ritualized to the point of losing its familial meaning or it is held hostage to manmade traditions like “church membership”. This should not be so. We are one people, part of the same one Body that is the Bride of Christ and we should welcome to the table all who confess Jesus Christ as Lord.
The Lord’s Table is not your meal. You aren’t the one who sent the invite and you aren’t the host so you and I don’t get to tell someone that they are not welcome because we have created an artificial barrier or a higher standard than the one that Jesus set forth and Paul affirmed. We should as His people be constantly striving to find ways to be in fellowship over the Table with other Christians more often, not seeking to exclude one another from this critically important act.