Different preachers and denominations explain these warnings differently, but most evangelical churches provide at least some word of warning. I remember preachers fencing the table strongly in the fundamentalist Baptist churches I grew up in. This is not just a Presbyterian thing.
I usually fence the table with three invitations based on 1 Corinthians 10 and 11:
- You may participate if you are United with Christ.
- You may participate if you are United with the Body of Christ.
- You may participate if you are Pursuing Unity within the Body of Christ.
United with Christ
The name Communion reflects two meanings: First it means communion with Christ. We get our word "communion" from the concept translated "participation" in this verse.
1 Corinthians 10:16 - The cup of blessing that we bless, is it not a participation in the blood of Christ? The bread that we break, is it not a participation in the body of Christ? (ESV)
The meal portrays, celebrates, and even practices our fellowship with Jesus Christ himself. So only those who are United with Christ should participate. This means that the meal is only for born-again believers. If your life is characterized by sin in a way that causes you and the church to question whether or not you are born again, then you should not participate.
United with the Body of Christ
This communion is also with one another in the visible Body of Christ, the Church. Some pastors present this stipulation as "you must be a member of a Gospel-preaching, Bible-believing church." I often do that myself. Membership is the main way modern churches practice being united with the body of Christ. But since not all churches have membership, and since some sincere believers may be temporarily out of official membership for good reason, I prefer to state this in looser terms. What this at least means is that one cannot disdain the visible body of Christ, and refuse to participate in the visible body of Christ, and be welcome to participate in what communion represents. We get this idea from the following verse:
1 Corinthians 10:17 - Because there is one bread, we who are many are one body, for we all partake of the one bread. (ESV)
Pursuing Unity Within the Body of Christ
Lastly, communion with Christ and with the Body of Christ has very practical implications for what it means to participate in the body of Christ. At the least it means that you cannot behave in ways that refuse to demonstrate the unity the church is supposed to have in Christ. We get this idea from the following verses in context:
1 Corinthians 11:17-19 - But in the following instructions I do not commend you, because when you come together it is not for the better but for the worse.  For, in the first place, when you come together as a church, I hear that there are divisions among you. And I believe it in part,  for there must be factions among you in order that those who are genuine among you may be recognized. (ESV)Because of this controversial behavior, Paul gives very, very strong warnings:
1 Corinthians 11:27-30 - Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty concerning the body and blood of the Lord.  Let a person examine himself, then, and so eat of the bread and drink of the cup.  For anyone who eats and drinks without discerning the body eats and drinks judgment on himself.  That is why many of you are weak and ill, and some have died. (ESV)
These invitations and warnings are the reasons communion is withheld from those who have not yet made a profession of faith. They explain why churches take control over who can participate and who cannot participate. They also explains why restriction-from-communion and excommunication (out of communion) are steps in church-discipline (although aspects of this practice are left-over from the old parochial church eras).
If it is spiritually and physically dangerous to participate in an unworthy fashion, it makes sense for shepherds to guide and protect their parish. Shepherds want to guide the flock into the benefits and blessings of communion and protect the parish from the dangers of communion.