When you try and lift the chair up, you're really doing something else.
You're just clamping your butt down firmly on the seat. The force you apply upwards on the chair is causing a reaction in your arms, which actually pushes your body downwards. If you remain still, the system is in static equilibrium; and the centre of mass remains the same.
In your situation where you "jump off the chair, but take the chair with you"; the main point is that you jump off the chair to raise your centre of mass (since we have internal energy in our bodies we can produce the energy for that, and in fact, it raises the centre of mass of the man-chair system). You then apply a force from your hands upwards. The reaction force still pulls you downwards. The difference is, since you are no longer firmly planted on the chair; this downwards motion isn't resisted by the chair (and thus required from your hands, leading to infinite force required). Instead, the force you apply moves your mass down, and the chairs mass up (keeping the centre of mass the same, until you begin to fall due to gravity; turn the potential energy to kinetic energy, and dissipate that on the ground).
You don't need to even "jump" on your chair to test this. Just crouch low in the chair and then bring your upper body up quickly while trying to pull the chair up. The chair will actually be able to make a (small) jump.