What will happen to the equilibrium between carbonic acid and bicarbonate if I add a strong base to this system?

The equilibrium will shift in forward direction.

Explanation:

The equilibrium between the carbonic acid and bicarbonate ion is shown below as:

H₂CO₃ (aq) ⇔ H⁺ (aq) + HCO₃²⁻ (aq)

According to Le Chatelier's Principle, the change in any state of the equilibrium say temperature, volume, pressure, or the concentration, the equilibrium will oppose these changes and will shift in such a way that the effect cause must be nullified.

If a strong base is added to the equilibrium, the base will accept hydrogen ions which are formed in the right side of the equilibrium. Thus, there will be less hydrogen ions present and to compensate this effect, the equilibrium will shift in forward direction.