In the Church, there are three beliefs regarding who should be allowed to take communion: Open communion, close communion, and closed communion. Closed communion is the most restrictive of the three, permitting only members of the specific church offering communion to partake. Close communion allows all members of the denomination, regardless of immediate church membership. Open communion is the most permissive, permitting, and even in some cases inviting, all professed believers to partake.
Of the three attitudes, open communion is the one most supported by Scripture. Open communion allows for the individual fulfillment of Christ's command "Do this in remembrance of me". This is a positive command to the Church, and so already, by preventing professed believers from fulfilling it, we might actually be sinning ourselves. I feel like Mark 9:38-40, while not directly applicable, may give some guidance in this scenario: "Now John answered Him, saying, “Teacher, we saw someone who does not follow us casting out demons in Your name, and we forbade him because he does not follow us.” But Jesus said, “Do not forbid him, for no one who works a miracle in My name can soon afterward speak evil of Me. For he who is not against us is on our side."
Here we see that the disciples saw a non-disciple using Jesus' name to cast out demons. They tried to forbid him, but Jesus said no, let him, because if he's not against us, he's for us. I think the same principle can be used of the Lord's Table. No congregation owns the Lord's Table. We don't get to say "Lord, we saw someone who does not follow our congregation/denomination trying to take communion, but we forbade him because he is not a member of our denomination." We just don't have that right.
In truth, no church will ever be able to see the heart, not even of its own members. My brethren, even if you are married, and you think you know even your spouse inside and out, you will never be able to see their heart in the depth that God does. You know your heart, God knows your heart, but your pastor, your elders, these people have no idea what's going on in there. Church membership? It's a bit of paper that says "this is a community I am a part of". A denomination? Personally, I contend that denominations are problematic anyway, because the Bible utterly forbids sectarianism (see 1 Corinthians 1:10-17; 3:1-4). In other words, even the motivation behind close and closed communion (that being ensuring only true Christians partake) is flawed. Why? Because it excludes true believers, while not necessarily forbidding false believers.
Biblically speaking, communion seems to be a very personal thing between an individual Christian and our God. We are to examine ourselves to see if we're actually taking communion properly. If you don't believe, you're handling the cup in an unworthy manner. If you're not in fellowship with the community of believers, you're eating the bread in an unworthy manner. If you're living in unrepentant sin, you're guilty with regard to the blood of Christ. But if you have a clear conscience, if you love the brethren, if you believe Jesus is the Son of God who died for your sins and rose again on the third day, you are worthy to partake in the Lord's Supper. Therefore, open communion is by far the superior model. Let God judge those who partake when they ought not.