It would appear that the specific priestly duty of manipulating blood at the altar (under the Mosaic Covenant) has been fulfilled and replaced by gospel ministry (under the New Covenant). This seems to be clear from 1 Corinthians 9:13-14, where the apostle Paul is defending his right to receive financial remuneration from preaching the gospel;
Don’t you know that those who work in the temple get their food from the temple, and those who serve at the altar share in what is offered on the altar? 14 In the same way, the Lord has commanded that those who preach the gospel should receive their living from the gospel. (1 Corinthians 9:13-14).
There are two parallels here. One is to do with stipend, i.e. the parallel between ‘eating food from the temple’ and ‘receiving a living from gospel ministry’. The other is to do with the nature of these respective ministries (altar-service and gospel-proclamation). Both are God-given mechanisms to effect forgiveness of sins. Under the Mosaic Covenant, blood sacrifice was the means of effecting the forgiveness of sins (anticipating Christ’s final sacrifice). But now that Christ’s perfect sacrifice has put away sin once and for all, there is no longer any need for blood sacrifice of any kind. Instead, sins are forgiven through the gospel. As the message of Christ’s atoning work is proclaimed, and as people hear it and believe it, sins are forgiven. Thus we see that “the work in the temple” finds typological fulfilment in “the preaching of the gospel”.