In this paper, we study the degrees of freedom (DoF) region and sum DoF of the time-selective$K$-user interference channel in the presence of intelligent reflecting surfaces (IRSs). We consider both active and passive IRSs. While both types of IRS can attenuate the amplitude and change the phase of a reflected electromagnetic wave, active IRSs are also capable of amplifying the wave. We derive inner and outer bounds for the DoF region and lower and upper bounds for the sum DoF of the$K$-user interference channel in the presence of an active IRS and prove that the maximum value of$K$for the sum DoF can be achieved if the number of IRS elements exceeds a certain finite value. The analysis framework developed for active IRSs forms the basis for our analysis of passive IRS-assisted systems. In particular, we present probabilistic inner and outer bounds for the DoF region and probabilistic lower and upper bounds for the sum DoF of the$K$-user interference channel in the presence of a passive IRS and prove that the lower bound for the sum DoF asymptotically approaches$K$as the number of IRS elements grows large.