Theorethicaly, it would be possible to have stable aeroplane with Cp ahead of Cg and tailplane generating upforce to balance wings' pitch-up momentum, provided every change of AoA leads to greater change of pitching momentum produced by tailplane than wings'. Whether such design can really be made (or was ever made), I have no idea.
On conventional design with tailplane producing downforce, its stall leads to rapid pitchdown with vertical dive easily and quickly attained, so it has to be designed to be immune to stall. On some specific designs, heavy ice accretion on tailplane's leading edge can compromise this. Non-FBW canards, such as VariEze, absolutely must have foreplanes that stall before mainplane, otherwise canards still producing lift combined with stalled wing would produce further pitchup - not quite desired effect when stalling.
Moving the CG aft by transferring fuel into tailplane reduces pitch-down momentum of the wings by reducing its arm. With lesser downforce from tail, less lift is required to balance it (decreasing a bit the stall speed), also trim drag is decreased. Downside is with shorter Cp to Cg arm, restoring pitch momentum with change of AoA is also smaller, making the aeroplane less stable - that's why it is desirable to empty trim tanks before approach.