I have a few suggestions based on my own experience, but there are probably a few ways to approach the problem.
I would recommend that you create a global variable for controlling the timing in the piece. It need only contain spacer rests, and potential line breaks (marked by \bar "", which will tell LilyPond that a line break is possible here with \cadenzaOn). You should identify _all_ the possible places LilyPond might break your piece up into lines (and then it will pick the best ones). At a glance it's not really clear to me what would work for your piece but I made an example just to show what I mean.
About bar lines: if you remove the engraver then you will not be able to add bar lines at the end. Although I'm not sure why you need this (since the cadenza mode doesn't print them) what I imagine you should do is simply remove the stencil at the beginning with \override Staff.BarLine.stencil = ##f, and then at the end when you need a double bar just revert the override with \revert Staff.BarLine.stencil.
Here is your code with my additions (including the very long lines, which is how it pasted into my editor):
\version "2.18.2"
%% This variable controls the timing and line breaks
global = {
\override Staff.TimeSignature.stencil = ##f
\cadenzaOn
% \bar "" will tell LilyPond that it's acceptable to break the line at a given point
s2. \bar "" s4 \bar "" s2. \bar "" s \bar "" s4 \bar "" s2. \bar "" s \bar "" s4 \bar "" s2. \bar "" s \bar "" s4 \bar "" s2. \bar ""
}
\header {
title = "Oremus"
tagline = ""
}
\new GregorianTranscriptionStaff
\with {
\remove "Time_signature_engraver"
}
{
aeh'4 bes' c'' deh'' ees'' f'' geh''
}
voiceA = {
\clef treble
\cadenzaOn
\accidentalStyle dodecaphonic
<c'' ees'''>2. <geh' f'''>4 <deh' bes''>2. <f' aeh''>2. <c'' deh'''>4 <geh' ees'''>2. <aeh c'''>2. <bes f''>4 <ees' geh''>2. <deh' aeh''>2. <c' f'''>4 <geh' bes''>2. \bar "|."
}
voiceB = {
\clef bass
\cadenzaOn
\accidentalStyle dodecaphonic
deh,8[ deh,8] ees,4 geh,8[ deh,8 bes,8] ees,4 geh,8[ geh,8 c,8] ees,4 geh,8[ f,8 bes,8] r4 r16 deh,8[ deh,8 deh,8] ees,4 deh,8[ bes,8 f,8 geh,8] ees,4 deh,8[ geh,8 geh,8 aeh,8 deh,8 geh,8] r4 r16 deh,8[ deh,8] ees,4 geh,8[ deh,8 bes,8] ees,4 geh,8[ geh,8 c,8] ees,4 geh,8[ f,8 bes,8] \bar "|."
}
\addlyrics {
Om- \skip 1 ah- \skip 1 mi- \skip 1 dhe- \skip 1 \skip 1 wa- \skip 1 hri- \skip 1 \skip 1 Om- \skip 1 ma- \skip 1 ni \skip 1 ped- \skip 1 \skip 1 me- \skip 1 hung- \skip 1 \skip 1 \skip 1 Om- \skip 1 ah- \skip 1 mi- \skip 1 dhe- \skip 1 \skip 1 wa- \skip 1 hri-
}
\score {
\new GrandStaff <<
\new Staff = voiceA <<
\global
\voiceA
>>
\new Staff = voiceB <<
\global
\voiceB
>>
>>
\layout { }
\midi { }
}