I was always told that I should cut off the black ending of a banana before eating, because it would be dangerous. Sources for this claim can for example be found here (though that source is actually rebuking the claim, but I was not able to find any other online sources). Is it indeed true that eating this black ending is unhealthy?
No, you needn't cut off the end of the banana unless it's rotten.
There are a two main reasons why bananas turn black.
Bananas keep on maturing even when they have been harvested. This happens through enzymes and oxydation and it's part of the normal life cycle of the fruit. In fact, a slightly over-ripe fruit is likely more nutritious because of higher sugar content. If your banana is simply ripening, you can eat it, it's oxidized and not rotten.
In ripening banana (Musa sp. [AAA group, Cavendish subgroup] cv Valery) fruit, the concentration of glycolytic intermediates increased in response to the rapid conversion of starch to sugars and CO2.
Fungi of the Colletotrichum family (and in particular Colletotrichum musae is the banana-specific one) are endemic on vegetables and cause rot. One well known effect is tip rot in bananas. Probably these fungi are not particularly toxic (this is my assumption), but the normal "avoid rotten food" advice applies.