Constant IAS means that Mach number, Mno, (and TAS) will increase; you could experience shock stall due to shock wave formations on the wings. Constant TAS will also cause Mno to rise, as the local speed of sound decreases with increasing altitude (up to the Tropopause). They use IAS up to "Changeover- / crossover point", after which Mno is used as reference. Usually around FL280-ish.
Climbing at constant IAS will require you to pitch down in order to increase TAS (to keep IAS constant because of decreasing density). This will give a lower climb gradient and is sometimes referred to as the kinetic energy factor.