While information theory has been introduced to characterize the fundamental limitations of control and filtering for a few decades, the existing information-theoretic methods are indirect and cumbersome for analyzing the limitations of continuous-time systems. To answer this challenge, we lift the information-theoretic analysis to continuous function spaces by the I-MMSE relationships. Continuous-time control and filtering systems are modeled into the additive Gaussian channels with and without feedback, and the total information rate is identified as a control and filtering trade-off metric and calculated from the estimation error of channel inputs. Fundamental constraints for this trade-off metric are first derived in a general setup and then used to capture the limitations of various control and filtering systems subject to linear and nonlinear plant models. For linear scenarios, we show that the total information rate quantifies the performance limits, such as the minimum entropy cost and the lowest achievable mean-square estimation error, in the time domain. For nonlinear systems, we provide a direct method to calculate and interpret the total information rate and its lower bound by the Stratonovich-Kushner equation.