Okay, it's actually pretty simple so here we go..
ATTENTION, THIS TUTORIAL NO LONGER WORKS. TUMBLR RECENTLY UPDATED HOW ICONS ARE UPLOADED, SO IT DOESN’T WORK ANYMORE. I’m sorry. /:
In order to make a gif as your tumblr icon you’ll need to download this add on (if you’re on chrome user) in order to see it. That being said, firefox, internet explorer, and opera users don’t need any add ons to see it. They have it easy. After you’ve done that, you’re going to need a .gif to .apng converter. I recommend using this one (for windows) or this one (for macs). Just click the green button to download it, and i promise you it’s safe ‘cause i use it. Alright, now that you’ve downloaded the converter, it’s time to make the gif you want to use as your icon. Remember, you have to make sure the size is 64x64 or it won’t work. This is important! After you’ve made your gif or picked up to use, just open up your converter, pick the input file and then hit convert! Super easy, right? I know, I just blew your mind. It happens.