Tumblelog doesn't allow you to host any videos on its site. It only permits you to link to videos which are actually stored on other servers (e.g. on Youtube). So the only way you can do what you want is to copy the Youtube videos to your own space on a video-hosting site.
Youtube videos arrive on your PC in FLV format but most (possibly all?) sites which allow you to upload videos don't permit you to send them in this format. So you need to capture the Youtube video, in FLV format, and convert it to a format which you can upload to your own space on a server. For appropriate software, see here:
http://www.theanswerbank.co.uk/Technology/Ques tion521195.html
Once you've captured the original Youtube video, in MPG or WMV format, you can then open your own account on a video hosting service (e.g. Youtube, Vimeo, Veoh, Dailymotion, etc) and upload the video. Then link to the video from your Tumblelog page.
This solution means breaching the copyright of the person who originally posted the video to Youtube (assuming that they held the copyright in the first place) but it's the only way that you can overcome the problem of dead links.
Chris