This issue is resolved in version 22.4.0.01 of Tumblr for Android.
Here's what the original announcement read:
We've identified an issue affecting native videos in the Tumblr app for Android. We're working to fix it up, and we'll update this page as soon as the fix is out.
In the meantime, if you're having trouble playing videos in the Android app, you can do one of the following:
- Tap anywhere outside of the play button. This should play the video.
- Change your media auto-play settings to "Only on Wi-Fi" or "Always auto-play."
Thanks for your patience 💖