But nothing is falling, all of the temperature records are rising.

I see what you're saying. I had taken the use to mean the situation is tumbling, not the temperatures. But upon a closer reading (of the title specifically) it seems a more reasonable interpretation of the word tumble is:

Climate records tumble,

The object of the verb 'tumble' is "climate records". That is, the climate records are tumbling. A tumbling record is one which has fallen over and been surpassed. So what they're saying by using the word "tumble" is: previous climate records have fallen over and been surpassed.

I do agree it's a weird word choice, but I don't think it's wrong or even playing on a potential uncommon secondary definition. It's not saying temperatures have tumbled, but rather records have tumbled.