A number of priests I’ve known know some or all of the Eucharistic Prayers by heart. I was noticing the other day my new pastor pretty much not glancing at the Sacramentary while praying the Roman Canon of all prayers. I’m pretty sure he knows it from memory.
One priest friend of mine related the story (about twenty years ago) that during a retreat/home Mass, somebody snatched his text away and said he was a good enough presider that he didn’t really need it. He didn’t, but I think it was an alarming experience.
Anyway, sit in the purple chair and advise your pastor: is it good to know all the EP’s without reading the text? Is it a distraction? Does it affect the prayerfulness of the priest?
Catholic Sensibility 
I’m confident that I know EP’s II and III and EP for Reconciliation I by heart, and yet whatever EP I’m praying, I have the book in front of me. I don’t very much look at the book at all although I turn the pages at the appropriate moments. If the book were not there, I’m not sure I’d feel as confident. I think the bottom line is this: with or without the book, memorized or not, does the presider pray the EP in a way that leads the assembly to prayer. I’ve seen many presiders speak the EP with little intonation, inflection, expression or prayerfulness and this is communicated to the assembly.
On a related note, I recently blogged about standing behind an aging parishioner who lip synched the entire liturgy along with the priest. Very impressive and prayerful, she was.
One of our vicars “does” what ConcordPastor illustrates. I find it edifying, actually.
I say don’t try it, you’ll only wind up forgetting it. I once witnessed a priest praying the collect for peace from memory – it was a trainwreck. However, I might suggest instead a priest should learn to SING the EPs from memory – I find it much easier to sing from memory than to speak. Or just learn to sing them period.
I know EP3 chanted by memory. Probably could get 2 and 3 by memory recited, but would never try. I’m with ConcordPastor on this one.
I know large amounts by heart, but I don’t rely on that; I get mixed up all the same for many reasons, and really, it’s not important that I have them memorized, but that I pray them.
As far as someone snatching the missal off the altar, such as you cited–woe betide anyone who does that to me!
In the Orthodox Typicon, which is the (very) big book of rubrics for the Orthodox services, there is a statement that no one, whether bishop, priest, deacon, server, cantor, or reader, is to rely on his memory alone, but is always to read from the appropriate book instead. I think that that is a wise idea.
While I have memorized just about all of the liturgical music which we pray in our little church, and most of the liturgy, I wouldn’t think of relying on my memory alone.
It depends on the Priest and his abilities. I’d suggest always having the text there in case humanity strikes.
One of my favorite mess-ups during Mass came from a Priest who was half following the book and it appeared half going from memory. Grabbing the chalice he said something like “Jesus took the cup in his hands and broke it, gave it to his disciples and sai…….err…”