Funerals can be in a Mass or not in a Mass.
If the funeral is in Mass, who should read the Gospel reading? If there is a deacon he should. If there are two priests, the one not giving the homily should. Only an ordained minister can proclaim the Gospel in a Mass.
If the funeral is in Mass, who is permitted to proclaim the first reading? From the General Instruction of the Roman Missal (G.I.R.M.) for the U.S.A. (at
usccb.org/prayer-and-worship/roman-missal/general-instruction-of-the-roman-missal/girm-chapter-3.cfm ):
“101. In the absence of an instituted lector, other lay people may be deputed to proclaim the readings from Sacred Scripture, people who are truly suited to carrying out this function and carefully prepared, so that by their hearing the readings from the sacred texts the faithful may conceive in their hearts a sweet and living affection for Sacred Scripture.[85]”.
So if there is a instituted lector, he should do the readings apart from the Gospel. If not, the choice of reader is up to the parish priest or rector of the church, from the G.I.R.M.:
“107. Liturgical functions that are not proper to the Priest or the Deacon and are mentioned above (nos. 100-106) may even be entrusted by means of a liturgical blessing or a temporary deputation to suitable lay persons chosen by the pastor or the rector of the church. [footnote 88: Cf. Pontifical Commission for the Interpretation of Legal Texts, response to dubium regarding can. 230 §2: Acta Apostolicae Sedis 86 (1994), p. 541.]”
Funerals can be outside a Mass, as described in the liturgical book Order of Christian Funerals, section 9. “303. … Depending upon pastoral circumstances, either one or two readings may be read before the gospel reading.” (The Rites Volume One, Liturgical Press, 1990, ISBN: 0-8146-6015-0, page 1030.) There are not the same explicit instructions about ordained ministers and instituted lectors. But it seems appropriate that what should happen in a Mass should also happen in this ceremony. If there is a priest, then a minister without ordination should not read the Gospel. If there is an instituted lector, then another person should not proclaim the first reading.