What is the English term for when someone thinks they are doing something nice for you but it ends up making things worse. EX: Someone buys you an elephant -- nice gesture and cool! But now you have to take care of it, and it becomes a burden on you.
In fact, the English expression for a burdensome gift is literally white elephant:
a thing that is useless and no longer needed, although it may have cost a lot of money [OALD]
So-called white elephants, or albino elephants, are found in many parts of South and Southeast Asia. In Buddhist countries they may be venerated as Queen Maya, mother of the Buddha, was said to have been visited in a dream by a white elephant holding a white lotus flower, and Siddharth Gautama entered his mother's womb in the form a white elephant. The white elephant is also associated with traits like mental strength and purity.
It became a royal symbol in Siam (Thailand); the king continues to keep white elephants. The story emerged that if a courtier displeased him, the king would make him a gift of a white elephant. The courtier could hardly decline a royal gift, and could hardly afford not to maintain a sacred animal, and could not put it to productive use, and so would be ruined by the cost of upkeep.
The earliest example of its use is from a 1721 essay in London Journal:
In short, Honour and Victory are generally no more than white Elephants; and for white Elephants the most destructive Wars have been often made.
A 2011 paper by Ross Bullen entitled “This Alarming Generosity”: White Elephants and the Logic of the Gift, in American Literature, covers the popularization of the term in the mid-19th century, presents an alternative account, that the story is a piece of orientalism and the white elephant rose as a literary trope.
This is perhaps darker than the question envisages, but I add it for its literary pedigree*. The Cambridge Advanced Learner’s Dictionary & Thesaurus explains it as:
“Something that seems very good when it is first received, but in fact does great harm to the person who receives it”
*According to Oxford Reference:
An assignment, award, or honour which is likely to prove a disadvantage or source of problems to the recipient; the phrase is found originally in Shakespeare’s Macbeth (1606), in a speech in which Macbeth flinches from the prospective murder of Duncan.
"White elephant" refers to a gift that's a burden, but the origins came from a situation where the burden was intentional. Albatross, money pit, and millstone all refer to burdens, but not specifically gifts.
a gift that keeps on giving
An ironic expression that indicates the "gift" has some ongoing and undesired consequences.
Perhaps a stretch, but I offer Pandora's Box:
Zonk was popularized by the U.S. television game show Let's Make a Deal. From the Wikipedia article:
The [show's] format ... [traders] make deals with the host. In most cases, a trader will be offered something of value and given a choice of whether to keep it or exchange it for a different item. The program's defining game mechanism is that the other item is hidden from the trader until that choice is made. The trader thus does not know if he or she is getting something of greater value or a prize that is referred to as a "zonk," an item purposely chosen to be of little or no value to the trader.
Emphasis is mine. Any connoisseur of U.S. television game shows will know this.