In one of my Azure Web App Web API application, I am creating temp files using this code in a Get method
string path = Path.GetTempFileName();
// d
Those files only get cleaned when your site is restarted.
If your site is running in Free or Shared mode, it only gets 300MB for temp files, so you could run out if you don't clean up.
If your site is in Basic or Standard mode, then there is significantly more space (around 200GB!). So you could probably get away with not cleaning up without running into the limit. Eventually, your site will get restarted (e.g. during platform upgrade), so things will get cleaned up.
See this page for some additional detail on this topic.