The reactions of H-3(+) with CO and with 0(3 P) are the two most important reactions for the destruction of H-3(+) in dense interstellar clouds. These two reactions are studied with sophisticated theoretical methods that should provide accurate predictions for the rate coefficients. The potential energy surfaces are studied with high-level electronic structure methods. For both reactions, simple long-range expansions are shown to be sufficiently accurate for predicting the kinetics at room temperature and lower. The kinetics is predicted from a combination of transition state theory, trajectory simulations, and master equation analysis. For the O(P-3) reaction, the interplay between the spin-orbit and the charge-quadrupole interactions is explicitly considered. For the CO reaction, we also consider the isomerization and decomposition dynamics of the two initially formed adducts. The final predictions, which are expected to be accurate to about 10 to 20%, are compared with the available experimental data. For the O(P-3) reaction, the predicted rate coefficient is accurately reproduced by the expression 1.14 x 10(-9) (T/300)(-0.156) exp(-1.41/T) cm(3) molecule(-1) s(-1) over the 5 to 400 K temperature range. For the CO reaction, the predicted rate coefficients for the H-2 + HCO+ and H-2 + HOC+ channels are accurately reproduced by the expressions 1.36 x 10(-9) (T/300)(-0.142) exp(3.41/T) and 8.49 x 10(-10) (T/300)(0.0661) exp(-5.21/T) cm(3) molecule(-1) s(-1), respectively, over the 10 to 400 K temperature range. These revised rate coefficient expressions imply an increase in the destruction of H-3(+) at temperatures that are typical of dense clouds (10-30 K) by a factor of 2.5 to 3.0.