Remote sensing images have found numerous applications nowadays. A traditional outcome or intermediate result of their processing is a classification map. Such maps are usually obtained from a pre-trained classifier and it is desired to have the produced classification maps as accurately as possible. The basic subject of this article is the factors determining this accuracy. The main among them are the quality of remote sensing data and classifier type, parameters and training approach. Image quality can be degraded due to several factors. One of them is distortions introduced by lossy compression that is widely used due to a huge volume of acquired data and the necessity to sufficiently decrease their size at transmission, storage and/or dissemination stages. Because of this, the main goal of this paper is to consider classification and lossy compression jointly. In particular, this means that the classifier learning can be performed for original (uncompressed, compressed in a lossless manner) images (if they are available) as well as for compressed data at hand (offered to a user for classification and further analysis). The task of this paper is to consider and compare these two options. The first one is the classifier learning for original images and further application to compressed data, where images can be compressed with different compression ratios while producing compressed data of different quality. The second option is the use of the classifier learning for compressed images, where compression parameters for training data can be approximately the same as for the images to which the classifier is applied. The main result is that the latter methodology can provide certain benefits compared to the classifier learning for original data if one has to classify compressed remote sensing data. Simulation data are obtained for a classifier based on a convolutional neural network. As images for training and verification, four real-life three-channel (visible range) Sentinel-2 remote sensing images of Kharkiv and Kharkiv region are employed that possess different complexity of the content and have four main classes. The practical recommendations are given. In conclusion, we can state that it is worth having classifiers trained for several degrees of compression and it is reasonable to compress complex structure images with special care.