Quantum measurement back action is fundamentally unavoidable when manipulating electron spins. Here we demonstrate that this back action can be efficiently exploited to tune the spin relaxation of localized electrons induced by the hyperfine interaction. In optical pump-probe experiments, powerful probe pulses suppress the spin relaxation of electrons on Si donors in an InGaAs epilayer due to the quantum Zeno effect. By contrast, an increase of the probe power leads to a speed-up of the spin relaxation for electrons in InGaAs quantum dots due to the quantum anti-Zeno effect. The microscopic description shows that the transition between the two regimes occurs when the spin dephasing time is comparable to the probe pulse repetition period.