Personally, the moment you substitute a minor tonic chord for a major tonic chord, you start slipping into the respective minor key. Change all your tonic chords to minor chords and I strongly doubt your piece is in a major key, even if all the other chords are pulled from the tonic major key.
I've heard some pieces with both major and minor chords for the tonic that I remained convinced was in the tonic major key. One example is the Circuit theme from Mario Kart Double Dash, which sometimes uses a C minor chord for the tonic but sticks to C major chords in important cadences. That game's title theme uses D major and minor chords in a similar fashion. I believe I've heard old pop songs that pull off something similar.
But that's the thing. They still use major tonic chords at the points that matter most--the cadences. Make all of those tonic chords minor and your piece won't really sound like it's in a major key anymore.
And another thing: the characteristic sound of modes depends precisely on their blend of major/minor tonic chords with minor/major/weirder accompanying chords. Without the constant use of chords like i, IV, (b)VII, and v, a piece wouldn't sound like it's in the Dorian mode. (For instance, swap out the IV chords for iv ones and you're in the Aeolian mode instead.)