When in a situation in which people may choose Scripture readings–and here I speak primarily of weddings and funerals–most priests and liturgists I know give the grieving family or the engaged couple pretty wide latitude.
These days, we rarely get people agitating for a non-Scripture passage. Sometimes, they will ask about a non-Lectionary reading. One of the favorites over the years is Ecclesiastes 3:1-11 (There is a time for this, and a time for that …) for the funeral.
My current boss has been known to let the people have two OT or two NT readings if they want. Twice in the past two years I’ve prepared Psalm 23 as a “reading” for a family member.
I know one priest who always reserves the choice of a funeral Gospel reading to himself. He invariably chooses the people’s favorite, John 14:1-6. Another priest friend once questioned me about the choices he encountered. Why, he asked, do people pick the Beatitudes for the funeral? Why do we allow Ecclesiastes?
Have a seat in the Chair of Choice. When the reading choices are significant, do you let the laity choose? Do you reserve any choice for yourself? Do you permit non-Lectionary Bible choices?
Catholic Sensibility 
We offer families helping to prepare a funeral a copy of the lectionary choices but we allow virtually any scripture text they choose, providing only one is from the gospels.
People choose readings in these circumstances for a variety of reasons, often reasons not rooted in sound exegesis, but if one or another text speaks to them, reaches them, touches them then it seems to me that overrides the impulse to make of the funeral an exercise in scripture study.
Ecclesiastes is a VERY popular choice here and while I don’t find it the best one, I understand how folks hear it, what they find in it, and that it’s a text they can grab on to for support in their grief.
When a family asks to use Psalm 23 as a lection, we assure them that it will be used as the sung psalm (offering them several musical settings) but we don’t use it as a reading.
While families are given the option of choosing the gospel, many do not (this may be because they are reminded that the priest/deacon will read the gospel). In light of that, my own practice is to use John 14:1-6 almost invariably because it announces beautifully a theme that is present in just about every homily I preach at a funeral.
I follow this same path with preparing for weddings where it’s my experience that many couples find the choices limiting. With the same cautions noted above, I invite them to consider non-Lectionary scriptural choices – and that invitation usually sends them back to rite’s list of readings.
I find that most funeral families have little difficulty making a selection from both the Hebrew and Christian scriptures but that some wedding couples do. In such cases, I permit two readings from the Christian scriptures or tell them that one reading is sufficient.
Please excuse the sequential comments here.
Just wanted to say thanks, Todd, for opening up a question like this. I think there are many instances in which pastors make choices like these. These aren’t radical, off the charts exceptions to the rules but simply playing with the given pieces to make the puzzle pastorally effective.
Well, the instructions for the current rite of matrimony were issued in 1969 and thus are somewhat vague in certain respects that perhaps now we’d like more granularity for. It does provide a list of lections that may be used. How restrictive that “may” (at least as translated into English) is intended to be understood is not crystalline.
The order of Christian funerals, however, has at instruction 23 an interesting general prohibition: no substituting non-biblical readings for biblical readings. And that, I think, is very correct – if we are to take seriously the idea that Christ is truly present in the proclamation of Scripture.
So, no non-biblical readings during the Liturgy of the Word. If someone insists on a non-biblical reading: (1) put in a program or handout, or (2) read it (a) at the vigil, (b) before the beginning of liturgy, (c) during the personal remembrance,(d) after committal, or (e) at the collation, if any.
I am inclined to say that, to the extent the ritual books indicate pastoral need is to play a decisive role in the selection of readings, then that general rule would probably be the interpretive guide for understanding that the lections provided in the ritual are not necessarily exclusive of any others from Scripture, other rules provided in the ritual books (like, if there are three readings and one is from the OT, that one is read first) are followed.
I would say this is especially true of the responsorial psalm (or gradual), given how flexible the Church is regarding the choice of psalms generally. (I should also note, the Sequence “Dies Irae” is also permitted, as has been explained elsewhere.)
Given that the homily will be based on the readings chosen, the celebrant’s homiletic needs are also an important consideration, as noted in the ritual books.
All that said, it would be important to caution that it would be inappropriate to criticize, as unreasonable or wrong, a celebrant who required readings to be selected from among those listed in the ritual books.