Through open market operations, the central bank may buy government debt to increase the money supply. Does the government need to pay interest on the debt held by the central bank, or does the debt become interest-free?
Yes, the government has to pay interest on debt even if it is held by the central bank. That is true at least for major modern central banks.
For example, from a press release from the US Federal Reserve Board:
Net income for 2020 was derived primarily from $100 billion in interest income on securities acquired through open market operations...
However, although modern central banks are in their decision making independent from the government, they are still public entities (or equivalent to such), and profits (if there are any) will ultimately be transferred back to the government. So, to the extent that profits are due to the holding of government debt, the government will get a rebate on interest paid.
This may vary from country to country, but in most countries government pays interest de jure but not de facto as all major central banks send their profits (including profits made from interest payments) back to their governments as they are government institutions.
For example, take Fed. US has to, de jure, pay Fed interest on its bonds.
However, de facto the debt is free because Fed just sends all it's net profits to the US government. For example, in 2020 Fed sent all its 88.5 billion profits to the US government and it does so each year (see WSJ).
This is an equivalent of government setting up an organization with independent budget, purchasing something from that organization, but then forcing the government controlled organization to sent its profits back to the government. The books of government and central banks are kept separate, but de facto Fed is US controlled institution so if Fed makes profit US government takes that profit and if Fed purchases debt from US government, it is de facto US government funding itself and sending itself back interest payments.
The ridiculous part of the whole thing is, if the fed decides to "buy" bonds" and the hold over two trillion of them... they didn't pay a dime for them, they simple increase the bank balance of what ever bank they "bought" them from. But there is no corresponding decrease in anyone's account for the purchase. The fed's increase in a bank account of some other entity, has no offsetting withdrawal elsewhere.
And of course, when the fed receives interest on its bonds they turn it back over to the treasury! So in effect 2 trillion of the bonds pay no interest.
And I'm sure the Fed thinks it could simply not make an issue when the bonds they hold come due for payment. And I predict someday when things get totally out of hand, they will do just that.
They are playing a deadly "shell game of fraud and deceit", running around in meaningless circles, so no one, they think, gets the true picture of what is happening. And part of it is interest rates of .25% for overnight lending of money. In other words it doesn't cost money to borrow money.