Well if every note was the same note as a major scale, it wouldnt be a lydian.
and if you're worried about playing some note that isnt in a major scale, then you're saying you cant play any other modes except for the major scale mode.
If you played a D mixolydian over a piece in D, it will work fine, because a mixolydian has all the same notes as the major scale, but with a flattened seventh.
you could mostly get away with that.(see bit at end)
To get a backing for a piece to have modes played over it, you use the notes of the mode to construct the chords.
In C major the notes are C D E F G A B C. constructing the chords goes like this:
G...A....B...C....D...E...F
E....F...G....A...B....C...D
C...D...E.....F...G...A...B...C
That gives you Cmaj, Dmin, Emin, Fmaj, Gmaj, Amin,Bdim
Using C mixolydian, the scale is C D E F G A Bflat C. the chords are made up like this:
G...A...Bflat...C...D........E....F
E....F...G.......A....Bflat...C...D
C...D...E.......F.....G.......A...Bflat...C
As you can see, this changes three of the chords, and the resulting chords are:
Cmaj, Dmin, Edim, Fmaj, Gmin, Amin, Bflat maj
Now, when you are playing a mode over a backing for a major scale, it sometimes wont work.
In the case of an Aeolian mode, it is a natural minor, and won't work at all. You could get away with using the mixolydian mode over a major scale backing, if you avoided making clashes. In C mixolydian, again, if you played the flattened seventh on the root chord, you would make a C7 chord, which would sound nice, however if you played the Bflat over a G major chord, it would clash badly.
Hope I haven't confused you.