Question: It is well known that there is a strong connection between introducing more money into an economy and inflation, as well as removing money and deflation.

However, how does the economy as a whole "detect" if money is removed from it?

A similar question would be how it detects whether the supply of money is increased, but that I could understand and explain better as printing new money is regulated, so the government and banks know about the increase and act accordingly. (yes, I know, printing money is not the only way to produce it, but I would guess creating new debt is also regulated or measured in some way)

I could also explain extreme cases, where the amount of money created or destroyed makes up a significant portion of the total amount. If suddenly such a large amount of money appeared, people would start to buy more things, and as the supply of things to buy is not enough, people will pay more money for it to be able to get it by out-bidding others, so prices increase, and we have inflation. Or if a significant portion of the money supply is destroyed, there is an abundance of things to buy, but not enough money to buy things, so prices will have to be lowered otherwise nobody would buy anything, so we have deflation.

However, by what mechanisms does it happen that small (but not negligibly small) amounts of change in the money supply cause small amounts of inflation or deflation?

For example, if one very rich person would sell off some property in cash or take cash out of bank accounts, and just burn or bury that cash without telling anyone, how would the economy "know" it and by what mechanisms would it cause deflation? The amount in this example is not enough for people to consciously realize that money is missing from the economy so they could adapt their spending habits accordingly.

Answer: 
Money is no different from any other commodity in this regard. How does the economy "detect" that the quantity of oranges has changed to determine (in conjunction with the demand for oranges) the equilibrium price of oranges? In general equilibrium models the classic theoretical model was the tâtonnement or Walrasian auction.

> 
> A Walrasian auction, introduced by Léon Walras, is a type of
>  simultaneous auction where each agent calculates its demand for the
>  good at every possible price and submits this to an auctioneer. The
>  price is then set so that the total demand across all agents equals
>  the total amount of the good. Thus, a Walrasian auction perfectly
>  matches the supply and the demand.
> 
> 
> 

[Walrasian auction](https://en.wikipedia.org/wiki/Walrasian_auction) 

While Walrasian auctions are helpful for theoretical models and do exist in the real world(see for example [the Tokyo's Grain Exchange](http://www.sciencedirect.com/science/article/pii/S0378426616300504)), they are not the dominant mechanism for setting price changes. Competition among intermediaries seems vastly more common. In oranges, it might be the actions of wholesalers that attempting to pay the best price for farmer's inventories or supermarkets attempting to pay the best price for wholesaler inventories. If someone grows oranges for private consumption and never buys or sells any oranges then that production never gets detected and is not reflected in the price of oranges. 

With money it is more complex but fundamentally similar. Banks bid for money by setting the interest rate to attract funds. Firms bid for capital by offering to pay an interest rate to banks. Households decide how much money to hold and how much to invest or spend instead. If a miser has a billion in cash and burns it without telling anyone the market may indeed never find out.
To the question "how does the economy as a whole detect if money is removed from it?", i would answer: "with great difficulties and intermediate modifications, which, in the end, call the tune".
To deal with money, be it in macro or micro terms, is mandatory to avoid neglecting the difference between stocks and flows. That is: it has to be identified "existing" (or out-of-bank-system) and circulating money, both connected (of course) by a velocity.

Generally, we speak of "supply" of money as the size of aggregate-bank-system balance sheet. That is, the stock of out-there "existing" money.
What "the economy" "perceives", on the contrary, is the flow of money going to buy goods, services, and assets (neither to mix up, nor to neglect, the latter!)

Money does not appear, but is created, what is a slight and important difference, because it has a counterpart. Also, money cannot desappear, but is "cancelated", against a counterpart.

In your rich person example, the burnt notes are not distinguishable from hoarded ones for "the economy" (because it's a secret), thus counting as existing ("supplied") money, but not circulating. Central Bank, or in general the issuer of the notes, when making periodical inventories of "existing" notes (i ignore the procedure and frecuency of this), will note the lost ones, and will cancel the liability in its balance sheet (writting down a profit, that in macro terms cancels the lost of your millioner).