How to predict different data via neural network, which is trained on the data with 36x60 size?
Neural networks (insofar as I've encountered) have a fixed input shape, freedom permitted only to batch size. This (probably) goes for every amazing neural network you've ever seen. Don't be too afraid of reshaping your image with off-the-shelf sampling to the network's expected input size. Robust computer-vision networks are generally trained on augmented data; randomly scaled, skewed, and otherwise transformed in order to---among other things---broaden the network's ability to handle this unavoidable scaling situation.
There are caveats, of course. An input for prediction should be as similar to the dataset it was trained on as possible, which is to say that a model should be applied to the data for which it was designed. For example, consider an object detection network made for satellite applications. If that same network is then applied to drone imagery, the relative size of objects may be substantially larger than the objects for which the network (specifically its anchor-box sizes) was designed.  
Tl;dr: Assuming you're using the right network for the job, don't be afraid to scale your images/frames to fit the network's inputs.
I was training a neural network with images of an eye that are shaped 36x60. So I can only predict the result using a 36x60 image? But in my application I have a video stream, this stream is divided into frames, for each frame 68 points of landmarks are predicted. In the eye range, I can select the eye point, and using the 'boundingrect' function from OpenCV, it is very easy to get a cropped image. But this image has no form 36x60. What is the correct way to get 36x60 data that can be used for forecasting? Or how to use a neural network for data of another form?