KakaoTalk channel has three BizMessage types: AlimTalk, FriendTalk and ConsultationTalk.
AlimTalk and FriendTalk do not accept sending media message using the GIF format while ConsultationTalk does support GIF files as long as they are less than 5MB in size.
Sending an animated GIF is a little tricky as the animation will only start when the user opens the image in full. No animation will be shown in the chat window.
The KakaoTalk documentation contains more information about media messages.