The rapid development of deep-learning enabled task-oriented communications (TOC) significantly shifts the paradigm of wireless communications. However, the high computation demands, particularly in resource-constrained systems e.g., mobile phones and UAVs, make TOC challenging for many tasks. To address the problem, we propose a novel TOC method with two models: a static and a dynamic model. In the static model, we apply a neural network (NN) as a task-oriented encoder (TOE) when there is no computation budget constraint. The dynamic model is used when device computation resources are limited, and it uses dynamic NNs with multiple exits as the TOE. The dynamic model sorts input data by complexity with thresholds, allowing the efficient allocation of computation resources. Furthermore, we analyze the convergence of the proposed TOC methods and show that the model converges at rate O (1/√T) with an epoch of length T. Experimental results demonstrate that the static model outperforms baseline models in terms of transmitted dimensions, floating-point operations (FLOPs), and accuracy simultaneously. The dynamic model can further improve accuracy and computational demand, providing an improved solution for resource-constrained systems.