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 four types of IRSs, namely 1) active IRSs, which are able to amplify, attenuate, and add a phase shift to the received signal, 2) passive IRSs, which are able to attenuate and add a phase shift to the received signal, 3) passive lossless IRSs, which are only able to add a phase shift to the received signal, and 4) $\varepsilon$-relaxed passive lossless IRSs, which are able to scale the received signal by a number between $1-\varepsilon$ and $1$ in addition to adding a phase shift. 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 $K$ for the sum DoF can be achieved if the number of IRS elements exceeds a certain finite value. Then, we introduce 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. For the DoF analysis of passive lossless IRSs, first, we approximate it by the $\varepsilon$-relaxed passive lossless IRS and introduce a probabilistic lower bound for the corresponding sum DoF. We prove that this bound asymptotically tends to $K$. In addition, we define a relaxed type of DoF called $ρ$-limited DoF. We introduce a lower bound for the $ρ$-limited sum DoF of the passive lossless IRS-assisted $K$-user interference channel and prove that this lower bound asymptotically also tends to $K$.