No. Champions cannot die, and will only disappear from your Collection if you sacrifice them.
Champions can become stunned if they lose in PvP. A stunned status leaves them inactive for a certain length of time.
Champions will not be stunned if they lose a fight against the Undead. However, fighting the Undead depletes your Energy. If your Energy runs out you can wait for it to recharge by itself, or use Champion Stones to recharge it instantly.
An Undead’s HP does not replenish if you run out of Energy before the battle is over. You can finish it off later once your Energy has been recharged with the same Champion or a different one.