Björntjänst can be broken down into two words you might know fairly early on in your Swedish-learning journey.
Björn means 'bear', and the word has existed in almost the exact same form for centuries, probably coming from an older word for 'brown'. Then we have tjänst, which means 'service' and is used in a wide variety of contexts: a company or an app might offer various tjänster, kundtjänst is 'customer services', and a tjänsteman (literally 'service man') is a category of professionals, which originally meant providing services rather than products but these days usually refers to professions requiring tertiary education.
So what on earth could björntjänst mean?
It comes from a 17th-century fairy tale, The Bear and the Gardener by Jean de La Fontaine (the story itself probably has even older origins). In the story, a gardener befriends a bear, and entrusts the animal with several tasks including keeping flies away from the gardener while he takes his nap (tupplur). All is going well until one especially persistent fly appears, and the bear tries to crush it using a paving stone, killing the fly, but also the gardener. Hoppsan (oops), as the Swedes would say.
In the Swedish translation, the final line of this slightly bleak story is: Även om motivet är ädelt, vill man helst slippa en björntjänst! (Even if the motive is never so lofty, it's always best to avoid a bear-service).
So a björntjänst is a well-meaning action that has negative or even disastrous consequences. There's an equivalent term in Russian (medvezhya usluga) and German (der Bärendienst) and the French expression le pavé de l'ours (the bear's paving stone) refers to the same thing.
However, many Swedes aren't aware of this backstory, so you might hear björntjänst used in different contexts today.
For example, it might be used in the sense of 'a disservice' without the implication that the motives were good (in Swedish a more accurate translation would be otjänst), or it may even be misused to mean something like 'a big favour' by people who think the word björn is being used for emphasis.
One example of the latter came in 2015, when a municipal council put up posters advising residents to 'gör både dig och miljön en björntjänst' (literally: do both yourself and the environment a favour that backfires) while trying to advertise electric bikes and hybrid cars. You could even say that by creating these posters, the council did themselves a björntjänst.
Vi måste inte göra vårt land en björntjänst
We don't want to do our country a [well-intended] disservice
Om du pratar engelska med nyanlända som vill lära sig svenska, så gör du dem en björntjänst
If you speak English with new arrivals who want to learn Swedish, it's a well-intended action with negative consequences