Would it potentially have a big impact to use higher resolution images?
Yes, if you increase the input size to your convolutional neural network, the size of each activation map for each layer increases, so you will have more computation. Also if you use same architecture, the number of neurons and consequently the number of parameters, in dense layers increases.
Does it make sense and is it possible to use a pre-trained network on low resolution and re-training the last layer/classifier with higher resolutions?
The answer is no. When you train a network with a special size of input, you reserve variables to hold the weights and middle variables. If you increase the size of input, dense layers will have different size, so their number of wights should vary too.
To wrap up, for networks with classification tasks, it is appropriate to pass the network small size of images. For other tasks like edge detection where the information of edges can be destroyed by resizing, you have to be careful. In those cases you have to find an appropriate size of the image in order to keep the important information. The small size of the inputs is for reducing number of operations and number of parameters.