Herman Bavinck gives some biblical reasons why Reformed/Presbyterian confessional churches do not allow infants to partake of the Lord’s Supper. He gives quite a few different biblical reasons, but I want to highlight only a few of them (note – these are my numbers to make it easier to read, though Bavinck had more than these numbers and in different order).
1) In 1 Cor 10 & 11, Paul “leaves no other impression than that only self-conscious adult persons took part in the Supper.” In other words, partaking involves at least some cognition as to what the Supper is all about. In my terms, Paul’s words like “examine” and “discern” assume the participants are old enough to consider the main truths of Christ’s work for sinners. Regardless whether one thinks “discerning the body” means Christ or the church (though I believe the context of 1 Cor 11.29 demands soma refers to Christ’s body/blood), the participle (discerning) implies more than elementary cognitive skills. Young children can neither examine themselves nor discern Christ’s body.
2) “Withholding the Lord’s Supper from the children does not deprive them of any benefit of the covenant of grace.” This is huge. I’ve heard some say “Why excommunicate our kids from the covenant by not letting them partake of the covenant meal?” Answer: we baptize the children, says Bavinck, which is a sign and seal that they were born into the covenant of grace (1 Cor 7.14). “But things are different with the Lord’s Supper. Those who administer baptism to children but not the Lord’s Supper acknowledge that they are in the covenant and share in all its benefits. They merely withhold from them a particular manner in which the same benefits are signed and sealed, since this manner is not suited to their age. The Lord’s Supper, after all, does not confer a single benefit that is not by faith granted through the Word and through baptism.”
Above quotes from Bavinck’s Reformed Dogmatics IV.583-4.
Though not in explicitly covenantal terms, many early church fathers also interpreted Paul to be writing to those a bit older, those able to understand the Creed, repent of their sins, and renounce the devil (i.e. Justin Martyr, Clement of Alexandria, The Constitutions of the Holy Apostles, and so on). As another historical note, the Heidelberg Catechism (specifically in Q/A 81) and the Westminster Larger Catechism (Q/A 171) also take for granted that the recipient of the Supper is able to examine themselves, repent, and believe. For a much expanded discussion of this topic along the lines that Bavinck argues, see Cornelis Venema’s fine work, Children at the Lord’s Table?