The Term αἴρω
This term when employed to communicate the concept of משא attaches the nuance of lifting or raising a burden.
For example, consider Num. 4:15, 24, 47 (“the service of burden”); 11:12; Matt. 9:6; 11:29; 16:24; 27:32; Acts 4:24; Col. 2:14; 1 Jn. 3:5.
The Term ἀναφέρω
This term, to my knowledge, never translates the term משא when comparing the MT with the LXX; however, there is a conceptual parallel between the terms, and if considered, it attaches the nuance of carrying, bringing, the moving up from a lower position to a higher position of a burden, the offering of a sacrifice, or the bearing or taking up as a burden. In the NT and in the Apostolic Fathers, this mostly refers to the offering of sacrifices (literal or devotional) to God or to Christ offering himself up as the sacrifice for sins.
For example, consider possibly Isa. 53:12 (נשא); Mark 9:2; Heb. 7:27; James 2:21; 1 Pet. 2:24; Heb. 13:15; 2 Clement 2:2; Barnabas 12:7; Heb. 9:28; 1 Clement 16:12, 14.