The phrase a millstone round someone’s neck means a heavy and inescapable responsibility. For example, the following is from The Stage (London) of 21^st December 1995:

In that virtually unimaginable era before TV, radio was huge and you could become a big star by appearing on it. The audience and the wages were vast compared to today. […]
Those days are something of a millstone round the neck of today’s light entertainment producers. They work hard to create original, funny programmes but are constantly reminded of the Golden Age of Radio Comedy.

This phrase originated in the gospel of Matthew, 18:6; the disciples having asked Jesus who the greatest in the kingdom of heaven is, he places a little child among them and answers that whoever takes the lowly position of children is the greatest in the kingdom of heaven; he then says that whoever welcomes one such child in his name welcomes him, and:

Early Version (around 1382) of the Wycliffe Bible:
Forsothe `who shal sclaundre oon of these `smale leste, that byleeuen in me, it spedith to hym that a myln stoon of assis* be hanged in his neeke, and be drenchid in to the depnesse of the see.

(* Asses were often employed to turn millstones. Here, a millstone of asses translates Latin mola asinaria, meaning a millstone turned by an ass, in the Vulgate, in which the adjective asinarius/asinaria (from asinus/asini, meaning an ass) is a misunderstanding of a form of Latin as/assis meaning a pound-weight; the translation should therefore be a weighty millstone.)

King James Version – 1611:
But who so shall offend one of these little ones which beleeue in mee, it were better for him that a milstone were hanged about his necke, and that hee were drowned in the depth of the Sea.

One of the earliest instances of the phrase is from The History and Adventures of an Atom (London, 1769), by the Scottish author Tobias Smollett (1721-71):

He declared, that not a man should be sent to the continent, nor a subsidy granted to any greedy, mercenary, freebooting Tartar; and threatened, that if any corrupt minister should dare to form such a connexion, he would hang it about his neck, like a millstone, to sink him to perdition.

And, in Miscellaneous Dissertations on Marriage, Celibacy, Covetousness, Virtue, the modern System of Education, &c. (London, 1769), John Dove, a writer on moral philosophy, explained:

A man may not marry but under certain restrictions and limitations; if he ventures beyond those, his children are bastardized, his wife taken from him, himself ruined, and the officiating priest transported and hanged. The restraints on propagation by the marriage-acts, together with other political ones clogging the conjugal state, is a millstone on the neck of this kingdom; and if it be not taken off will eventually ruin it.