This might seem like a silly question, but I am not a Catholic, and so I do not know the answer!
Do Catholics tithe?
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Catholics are not required to give a tenth of their income.
From the Catechism of the Catholic Church:
1351 From the very beginning Christians have brought, along with the bread and wine for the Eucharist, gifts to share with those in need. This custom of the collection, ever appropriate, is inspired by the example of Christ who became poor to make us rich: Those who are well off, and who are also willing, give as each chooses. What is gathered is given to him who presides to assist orphans and widows, those whom illness or any other cause has deprived of resources, prisoners, immigrants and, in a word, all who are in need.
Of course, this doesn't mean they don't give. And Catholics are obligated to give, according to ability. This isn't just semantics, there's a difference between the two.
A tithe in this context isn't just an offering, it's a specific percentage of a person's salary that the person must give to their church, usually under pain of sin. This was required by the Old Law, but is no longer required. Charity (a special formal word that means love) obliges us to give, though. How the offering is gathered might depend on the particular parish, but it's roughly what you'd expect.
Sometimes, some of us even sing a song about this issue (Table of Plenty). Some of the lyrics go:
♫ O, come and eat without money. Come to drink without price.
My feast of gladness will feed your spirit with faith and fullness of life.
By "feast" we mean the Most Holy Eucharist, the Most Sacred Blood, God literally present before us, just as He was Himself present before St Thomas the doubter.
We sing that song being mindful of Isaiah 55:
1 All you who are thirsty,
come to the water!
You who have no money,
come, buy grain and eat;
Come, buy grain without money,
wine and milk without cost!
2 Why spend your money for what is not bread;
your wages for what does not satisfy?
Only listen to me, and you shall eat well,
you shall delight in rich fare.
However, again from the Catechism:
2043 [...] The faithful also have the duty of providing for the material needs of the Church, each according to his own abilities.
On the obligation to give, 2 Corinthians 9:
6 Consider this: whoever sows sparingly will also reap sparingly, and whoever sows bountifully will also reap bountifully.b 7 Each must do as already determined, without sadness or compulsion, for God loves a cheerful giver.
Addressing the Old Law, Romans 13:
8 Owe nothing to anyone, except to love one another;
for the one who loves another has fulfilled the law.
A person who has love for her Church will give an appropriate offering. In the same way that St John took care of Holy Mary, Mother of God, so too do we presently seek to take care of the Christ's Bride until she is reunited with Him at the end of time. In turn, she takes care of those who are in need (the Catholic Church is, among other things, a really, really big decentralized charity).
Well, I've been to Catholic services, and I have witnessed a basket being passed around wherein money is placed. If tithe is defined as putting money in that basket, then yes, they do.
Officially, that is, according to Roman Catholic doctrine, the Catechism, §2043, states
The fifth precept ("You shall help to provide for the needs of the Church") means that the faithful are obliged to assist with the material needs of the Church, each according to his own ability.
The faithful also have the duty of providing for the material needs of the Church, each according to his own abilities
Thus, the idea is that you are not commanded to give a certain percentage, although "tithe" literally refers to 10%. Rather, it is expected that one give according to his ability.
Catholics in Germany pay 8%-9% of their income tax to the catholic church. It's deducted directly from the paycheck like a tax and it's actually called Church Tax (Kirchensteuer). That seems to qualify as tithing. Now the church may claim that this is an artifact of German law and not their doing. However, they sure as heck take the money and I've certainly never see them refuse it because it wasn't given voluntarily. [EDIT: the tax is based on income tax and not taxable income, total revenue in 2012 was 5.8 billion Euro which is about 8 billion dollars]
Most of the answers above discuss the theology behind whether Catholics are supposed to. On the empirical side of whether people actually do, basically no. Many Catholics don't go to church that often and a large minority (maybe majority) live in fairly dire poverty, e.g. most Catholics in Latin America, Africa, and the Philippines.
A long time ago many Catholics were forced to tithe, e.g. peasants in the French semi-independent church (even as late as 1789), everyone who lived in the Papal States (I think).