Moderate or intense low-oxygen dilution (MILD) combustion of methane blended with hydrogen offers a potential solution for gas-fired industrial furnaces and even boilers to realize rapid decarbonization while maintaining ultra-low NO emissions. In this work, the combustion mode map for identifying MILD combustion when burning CH4/H2 mixtures from pure methane to pure hydrogen is proposed, and then the sustainability of hydrogen-enriched methane MILD combustion over a wide range of hydrogen-blending ratios from 0 to 100% under strongly heat-extracted conditions is revealed. Results show that, there exists a threshold oxygen mole fraction (XO2*) below which hydrogen-enriched methane MILD combustion can be achieved as long as the inlet temperature (Tin) is above the self-ignition temperature of reactants. As the hydrogen-blending ratio increases from 0 to 100%, XO2* is reduced from 9.3 to 6.5% and thus the unconditional MILD combustion region is concentrated, implying that hydrogen addition requires a larger oxygen dilution level to sustain MILD combustion. Interestingly, the sustainable operating range of oxygen concentration to sustain hydrogen-enriched methane MILD combustion can be extended by increasing the heat extraction ratio (HER) properly, e.g., XO2* up to 13.2% at HER = 30% for pure hydrogen fuel. However, methane MILD combustion would become unstable and even extinguished once HER exceeds about 49%, whereas MILD combustion of methane blended with a high percentage of hydrogen could be sustained even under highly heat-extracted conditions, e.g., a maximum allowable HER of about 91% for pure hydrogen fuel. In pure hydrogen MILD combustion, heat extraction can further reduce NO emissions, mainly due to less NO production via the N2O-intermediate, NNH, and thermal pathways, and NO reduction by H radicals tends to be weakened with heat extraction. In conclusion, high-percentage hydrogen co-firing makes it possible to achieve stable MILD operation with low NO emissions under strongly heat-extracted conditions encountered in boilers.