How much tax revenue did the British Empire receive from its colonies in a given year?

I will take the American Colonies. It's extremely difficult to quantify revenue because often Parliamentary Acts were passed which favored British industries at the expense of colonial industries, or transferred the cost of direct expenses individuals in the colonies rather than asking for revenue.

What Drove the British Taxation System in 1760s and 70s:

TaxHistory.org
The British Government had borrowed heavily from British and Dutch bankers to finance the (French and Indian, or Seven Years War), and as a consequence the national debt almost doubled from £75 million in 1754 to £133 million in 1763. In order to address this onerous liability, British officials turned to larger import duties on enumerated goods like sugar and tobacco, along with a series of high excise (sales) taxes on goods such as salt, beer, and spirits. This taxation strategy tended to burden consumers disproportionately. In addition, government bureaucracy expanded in order to collect the needed revenue. As the number of royal officials more than doubled, Parliament delegated new legal and administrative authority to them. Thus, even as British subjects lauded their preeminent position in the world, they chafed under the weight of increased debts and tightened government controls.

Given Britain’s exertions on the North American continent for the sake of colonial security, both ministers and members of Parliament determined that the colonies were obligated to share the costs of empire.

What the Colonies were thinking.

Taxation without Representation
Totally History
According to the British Constitution, one cannot take away the property or taxes of a British subject without representation in the Parliament. In this case, since the colonies were not represented in the Parliament, the colonies fought against the Parliaments’ devious acts, with some even saying the Parliament has no right to take away and use the taxes. This they clearly expressed in the slogan: No taxation without representation.”

How much taxes were taken depended upon which year and which Act. There were many different acts by the British Parliament which taxed Colonial America in many ways not all quantifiable.

Yes the British took tax revenue, or sometimes in conjunction with an act they prohibited a colonial industry in order to curtail competition with a native British Industry (aka the wool and steel acts). Or sometimes they would simple impose a monopoly for one British company over existing companies as a royal favor(1773 Tea Act). Some acts would require all ships coming in and out of the colonies to have at least half British Crews, or most goods to be transported on English ships (1651 Navigation Act). Or an Act which said British Troops could be Quartered and Fed in personal homes at personal expense. Such bills cost Colonial Americans money in the form of increased costs, and increased savings to the realm but could not necessarily be measured in revenue to the crown.

The below bullet items are taken directly from the following primary sources:
British Colonial Trade Regulations, 1651-1764
History of British Acts & Taxation in the Colonies

• 1651 Navigation Act
  • Required all crews to be at least 1/2 English in nationality
  • Most goods must be carried on English or colonial ships
  • Goal: eliminate Dutch competition from colonial trading routes
• 1660 Navigation Act
  • Required all colonial trade to be on English ships
  • Master and 3/4 of crew must be English
  • Long list of "enumerated goods" developed, including tobacco, sugar, rice, that could only be shipped to England or an English colony
• 1663 Staple Act
  • Required goods bound for the colonies shipped from Africa, Asia, or Europe to first be landed in England before shipping to America.
• 1673 Plantation Duty Act
  • Required colonial ship captains to guarantee that they would deliver enumerated goods to England or suffer financial penalties.
  • Colonial arm of English customs
• 1696 Navigation Act
  • Further tightened earlier Navigation Acts
  • Created system of admiralty courts to enforce trade regulations and punish smugglers
  • Customs officials given power to issue writs of assistance to board ships and search for smuggled goods
• 1699 Wool Act
  • To prevent competition with English producers, prohibited colonial export of woolen cloth.
• 1765 Stamp Act
  • Designed to raise revenue from the American Colonies by a duty (tax) in the form of a stamp required on all newspapers and legal or commercial documents.
  • The year the British Stamp Tax was passed 1765. The Stamp Tax called for the taxing of 54 separate items The Stamp Act
    1765 Guided by Prime Minister In order to cover about £60,000 of the £200,000 required to station troops in the colonies, George Grenville persuaded Parliament to pass a Stamp Act similar to one enacted and successfully administered in England in 1694.
• 1732 Debt Recovery Act
  • Appease British merchants who were owed money by the colonists
  • Allow creditors to seize property, land and slaves
  • Guarantee uniform imperial treatment of slave property for debt-recovery purposes in England and its colonies
  • Slaves were be treated as personal property and used to clear debts
  • Override existing colonial laws in relation to property
• 1773 Tea Act
  • The Tea Act imposed no new taxes
  • It gave a tea monopoly in the American colonies to the British East India Company
  • The Tea Act allowed the East India company to sell its large tea surplus below the prices charged by colonial competitors
• 1732 Hat Act
  • Prohibited export of colonial-produced hats.
• 1733 Molasses Act
  • All non-English imported molasses taxed heavily to encourage importation of British West Indian molasses.
• 1750 Iron Act
  • British Law, passed by the Parliament of Great Britain, that was designed to encourage the American manufacture of more pig iron and iron bars by the American colonists in the 13 Colonies to be sent to England, tax free. But the Act of 1750 also prohibited the colonies from producing finished iron goods.
• Proclamation of 1763
  • Halted the westward expansion by colonists whilst expanding the lucrative fur trade.
• 1764 Currency Act
  • Extend the provisions of the Currency Act of 1751 (New England Currency Act)
  • Control the printing and use of colonial paper money (Bills of Credit)
  • Appease British merchants who did not trust colonial paper money due to its depreciation in value. British merchants in England wanted to be paid in British currency and not colony currency
  • Reduce the national debt
• 1765 & 1774 Quartering Act
• 1764 Sugar Act(American Revenue Act)
  • Lord Grenville institutes new policies to generate revenue by combining new duties on imported goods with strict collectiion provisions. Tax on French West Indies molasses was actually lowered, but enforcement attempted to end bribes and smuggling.
• 1766 Declaratory Act
  • Stated that Parliament's authority was the same in America as in Britain and asserted Parliament's authority to pass laws that were binding on the American colonies. The colonies did not dispute the notion of Parliamentary supremacy over the law. But the ability to tax without representation was another matter. The Declaratory Act made no such distinction.
• 1767 Townshend Acts
  • A series of laws which set new import taxes on British goods including paint, paper, lead, glass and tea and used revenues to maintain British troops in America and to pay the salaries of some Royal officials who were appointed to work in the American colonies.
  • Townshend Act: The Revenue Act of 1767
  • Townshend Act: The Indemnity Act of 1767
  • Townshend Act: The Commissioners of Customs Act of 1767
  • Townshend Act: The Vice Admiralty Court Act of 1767
  • Townshend Act: The New York Restraining Act of 1768

US History.org
The revenue generated by the Townshend duties, in 1770, amounted to less than £21,000. On March 5, 1770, Parliament repealed the duties, except for the one on tea.

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And of coarse this tea tax was the one which inspired the Boston Tea Party, and the Boston Tea Party was what inspired the intolerable acts.

Wikipedia: Intolerable Acts.
- 1774 Coercive Intolerable Acts
- The Boston Port Act - The first of the Intolerable Acts closed the port of Boston so tightly that the colonists could not bring hay from Charlestown to give to their starving horses.
- The Massachusetts Government Act - The second of the Intolerable Acts put an end to the constitution of Massachusetts - only one town meeting was permitted a year in Massachusetts, unless approved by the governor. Town officials would no longer be elected, they were to be be appointed by the royal governor. The executive council would no longer be elected, but appointed by the King. The Massachusetts Government Act revoked the colony's 1691 charter.
- Administration of Justice ActThe third of the Intolerable Acts gave the power for all accused royal officials trials in the colony to be sent to Great Britain and heard under a British judge. George Washington called this the "Murder Act" because he believed that it allowed British officials to harass Americans and then escape justice.
- Quartering Act - The fourth of the punishing Intolerable Acts compelled the colonists to feed and shelter the soldiers employed to punish them.
- A fifth act is also included in the Intolerable Acts of 1774 and relates to Quebec and Ohio. This 'Intolerable Act' was also passed in 1774 and, although it was not directed at punishing the colony of Massachusetts for the Boston Tea Party, it was seen as a new model for an authoritarian British colonial administration and another threat to the independence of the colonies