Whenever I head to school or somewhere else, I sometimes like to render my images(even if they're not complete) at something crazy, like 10,000 samples at a high resolution with many light bounces. Will this damage my computer or GPU, or take up space?
If your computer has proper cooling and therefore doesn't overheat, it won't damage the GPU.
It will have an unfavorable impact on your electricity bill and carbon footprint though.
Whether a sample size that high is actually appropriate is another question. You probably don't need that many samples or there are computationally less expensive solutions to reduce noise, e.g. the denoiser.
The rendered image will take up disk space when you save it. How much it will use depends on the resolution, the image format and its settings. The higher the resolution, the larger the file. For an overview of image formats and their disk space usage, take a look at the last section of this answer.
You're not going to "damage" your GPU. What may happen is that if it is constantly over-clocked or over-used, it may wear out faster than a less-used GPU. Render however you wish, as long as your GPU can handle the amount of processing you're asking of it, then it will be fine. However, 10k samples does seem very excessive.