The liturgical books (such as the Rituale Romanum) and most rubrical authors presume the clergy and ministers are familiar with biretta etiquette, thus few give explicit instructions on its use within the particular ceremonies themselves, though some outline the basic principles.
In this regard, Callewaert in his Caeremoniale [available from Romanitas Press] has an excellent chapter (pp 15-16) that thoroughly deals with the governing principles for ministers and assisting clergy.
For a lesser but similar treatment in English, one can consult Very Rev. Laurence O'Connell's The Book of Ceremonies, pp 48-49 and Msgr. J. Lane's Notes on Some Ceremonies of the Roman Rite (Newman, 1961), pp 9-10.
But to summarize an answer to the specific questions:
During a non-Eucharistic procession held outdoors (except with a relic of the True Cross), the biretta is worn by all clergy. The exception though is for the inferior ministers (whom never wear a biretta) and any cleric bearing a saint's relic or sacred image (banner, statue, etc.).
If a bow or genuflection is required or if one comes into the presence of the Blessed Sacrament exposed, the biretta is removed and held in front of the chest.
The assisting clergy remove their birettas upon entering a church (the vested sacred ministers retain theirs though); they do the same if moving from place to place during the ceremonies.
In an attempt to provide references concerning the burial service, a quick check was made through a few of the most notable English-language manuals (Fortescue, L. O'Connell, Baltimore Ceremonial, etc.). But this yielded only one specific mention on the subject in Mueller-Ellis' Handbook of Ceremonies (Herder, 1954). On p 232 it instructs all to remove their birettas upon reaching the grave. Hence the priest should not be wearing the biretta when giving the final blessing at the grave (as often seen in some older pictures).
Once again, we can rely on Callewaert for a detailed description about this point on p 193, ff 7:
Usus bireti in Absolutione et exsequiis: 1) dum proceditur ab ecclesia ad domum defuncti aut inde revertitur, aut post missam ab altari ad feretrum; a feretro ad sepulchrum et a sepulchro (aut feretro) ad sacristiam, omnes clerici, exceptis crucigero et ministris inferioribus, deferunt biretum extra ecclesiam, sed soli Celebrans et ministri parati (h.e. induti paramentis nigris, ad minus stola) servant biretum in ecclesia; 2) dum autem actu peragitur alique functio in domo defuncti (necnon dum cadaver e domo effertur), ad tumulum, ad sepulchrum et in fine in sacristia, omnes etiam Celebrans caput detegunt (nn. 27; 28).
Here is a rough translation:
The use of the biretta for the Absolution and funerals: 1) while proceeding from the church to the house of the deceased, or thence returning, or after Mass from the altar to the coffin, with the coffin to the grave and from the grave (or coffin) to the sacristy, all the clergy, and ministers excluding the crossbearer and inferior ministers, wear a biretta outside the church, but only the vested celebrant and ministers (that is, dressed in black vestments, stole at least) maintain the biretta in the church, 2) however while the actual function is performed in the house of the deceased person (as well as when the body is carried out), at the covering*, at the grave, and in the sacristy at the end, of whom all, including the head of the Celebrant, is uncovered (nos. 27, 28).
*a type of pall, or floor covering, placed on the floor if there is no coffin—today it is more common to use a catafalque (empty coffin) for this purpose.
Louis J. Tofari
Recently, in connection with a Peregrinus Gasolinus chapter (The Use of the Biretta), a priest made an inquiry concerning when the biretta may be worn in non-Eucharistic processions and during the funerary rites. Since this brings up some interesting points, I thought it would be a good idea to reproduce the response.