Q. Paul & Barnabas traveled and preached together, but split due to differing viewpoints on whether or not to take John Mark with them (Acts 15:36-40). Did they reconcile, or did they remain estranged?
A. There is no explicit scripture that details reconciliation between Paul and Barnabas. I believe, however, that there is a bit of evidence that indicates that Paul and Barnabas did not continue at odds with one another.
In 1 Corinthians 9 Paul makes a statement about his apostleship and his rights as an apostle. Within this statement, 1 Corinthians 9:3-6, Paul mentions Barnabas as a Christian laborer who also “forebears working” (v. 6). Or I only and Barnabas, have not we power to forbear working? What is said is short and straightforward. Paul gives no broad discussion of his relationship with Barnabas. What little he says is nevertheless revealing. Paul recognizes Barnabas as a co-laborer. He holds him forth as an example as one who performs like ministry in a like way. There is no indication that he harbors any grudge. A restoration of fellowship can be easily assumed.
If we follow a biblical timeline, we find that this was written after the the split between Paul and Barnabas. The rift is recorded in Acts 15. From that point in time, Barnabas went with Mark and Paul went with Silas. Paul’s trip to Corinth is not recorded until Acts 18. This is during the time that Paul was laboring with Silas. The church began at Corinth is after the Acts 15 rift. Clearly, then, Paul wrote the first epistle to the church of Corinth after his rift with Barnabas. When he mentions Barnabas in 1 Corinthians 9, this mention occurs after the split of Acts 15. The favorable mention at this late date suggests that they had made their peace.
There is a more direct mention of Paul’s reconciliation with Mark, found in 2 Timothy 4:11 -- Only Luke is with me. Take Mark, and bring him with thee: for he is profitable to me for the ministry. It is unlikely that Paul reconciled with the “junior partner” (Mark) without having reconciled with the “senior partner” (Barnabas).
Other than this, I am unaware of anything else in the New Testament that speaks to the reconciliation of Paul and Barnabas. These two scriptures are enough to satisfy me. Your mileage may vary.