In intelligent reflecting surface (IRS) assisted communication systems, the acquisition of channel state information (CSI) is a crucial impediment for achieving the beamforming gain of IRS because of the considerable overhead required for channel estimation. Specifically, under the current beamforming design for IRS-assisted communications, $KMN+KM$ channel coefficients should be estimated, where $K$, $N$ and $M$ denote the numbers of users, IRS reflecting elements, and antennas at the base station (BS), respectively. To accurately estimate such a large number of channel coefficients within a short time interval, we propose a novel three-phase pilot-based channel estimation framework in this paper for IRS-assisted uplink multiuser communications. Under this framework, we analytically prove that a time duration consisting of $K+N+\max(K-1,\lceil (K-1)N/M \rceil)$ pilot symbols is sufficient for the BS to perfectly recover all the $KMN+KM$ channel coefficients for the case without receiver noise at the BS. In contrast to the channel estimation for conventional uplink communications without IRS where the minimum channel estimation time is independent of the number of receive antennas at the BS, our result reveals the crucial role of massive MIMO (multiple-input multiple-output) in reducing the channel estimation time for IRS-assisted communications. Further, for the case with receiver noise, the user pilot sequences, IRS reflecting coefficients, and BS linear minimum mean-squared error (LMMSE) channel estimators are characterized in closed-form, and the corresponding estimation mean-squared error (MSE) is quantified.