Both testaments of the Bible bear witness to the problem of competing religious ideas that were contrary to biblcal faith and theology in the Old Testament period, the polytheistic religious systems of the surrounding nations were a persistent threat to Israel's exclusive loyalty to YHWH as the true God. That loyalty was the core of Israel's faith (Deut. 6:4) and the basis for salvation in the Old Testament.
In the New Testament period, idolatry and the danger of worshipping a lesser god was still a targer of apostolic teaching and writing. Paul admonished the Corinthians to be careful about eating meat sacrificed to idol, warning them that in doing so they were at risk of committing idolatry and fellowshipping with demons (1 Cor, 10: 14-22). Since Paul elsewhere permits eating such meat (1 Cor. 8:4, 7-9), advising believers not to make a fuss about offering that were sold in the marketplace (1 Cor. 10:25), the issue for Paul must have been avoiding any sort of activity that either was participation in the ritual or could bne construed that way.
The greater problem of false teaching, however, seems to involve teachers that professed to follow Jesus but whose teachings altered the simplicity of the gospel - that salvation was only by grace through faith (Eph. 2: 8-9) and was available to Jews and gentiles alike.
Paul and the other apostles repeatedly has to deal with "Judaizers," who argued that gentles had to conform to certian points of the Mosiac law in order to be in right standing before God. Paul referred to those who taught such things as "the circumcision party" and famously confronted Peter for his failure to oppose their teaching (Gal. 2: 11-14). Peter knew better but was afraid to speak out against Judaizing teaching at Antioch. Paul referred to his behavior as "hypocrisy" that was "out of step with the gospel" (Gal 2: 13-14). The Judiazing seems to have been focused in circumcision (and therefore Jewish identity) since Paul later accuses the Judaizers themselves of being lawbreakers (Gal. 6:13). The point is that they had focused on outward conformity to Jewish identity and not the heart. They presumed that being Jewish meant superior status with God.
Peter himself would later have a problem with false teaching. He, along with Jude, had to confront self-proclaimed prophets who rejected moral authority (Jude 4, 8-10). These false teachers were apparently professing Christians since they participated in the Lord's Supper (Jude 12). Jude rejected their profession, charging they didn't have the Holy Spirit (Jude 19).
As the Epistle show us, we shouldn't tolerate teaching that alters the gospel or encourages a life devoid of biblical morals. When it comes to false teaching, tolerance is no virtue.