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The idiom foot the bill means:
1. to pay all the costs for something (We ended up having to foot the bill for a new roof because our insurance didn't cover storm damage.) The Free Dictionary
2. to pay money owed; to pay for something (Who's going to foot the bill for all the repairs?) (often + for) The Free Dictionary
3. pay the bill, settle the accounts, as in The bride's father was resigned to footing the bill for the wedding. This expression uses foot in the sense of "add up and put the total at the foot, or bottom, of an account." [Colloquial; early 19th century] Answers
It comes from the mildly figurative sense of foot that refers to the end or bottom of something, such as the foot of a ladder...our sense refers in particular to the totting up of a column of figures, especially in an account ledger, and adding the result to the bottom of the column.
It often now has the implication of paying for something whose cost is considered large or unreasonable, especially if the person doing so has been forced into paying for the consequences of the actions of somebody else.
This modern-day usage is often how it is found in many articles and other current works, although many continue to use it with its more broad definition of simply to pay the costs or money owed for something, holding true its accounting origins.
Michael Quinion further states that the phrase was...
...often used in the set phrase foot up (to), meaning to count...“The united debts of the colony foot up something like £50,000”...“The two counted the pile and found it footed up to two hundred and forty dollars.” Our sense of settling one’s account was acquired...because adding up the items on an account was something that would commonly be done at the point when one was paying one’s bill.
To start with, it was a decidedly colloquial usage, but as time passed the associated senses fell out of use and to foot the bill is now a fixed phrase, though still somewhat informal.
As time goes by, often sayings fall out of favor (foot up (to)) while some continue on (foot the bill), despite ambiguous origins.
"Flip the bill" is an Eggcorn of this idiom; replacing the obscure "foot" with the more familiar "flip" with the help of a modern-day social norm.
A tradition in dining establishments (i.e. restaurants, diners, etc.) involves the server presenting a check to a party face down on the table. Hard pressed to find a website stating exactly what started this tradition leaves one to simply cite examples from daily life. For instance, in literature; this from an excerpt from Lee Child's "Echo Burning" :
She held the water glass flat against her face. Then she used a napkin to wipe the dew away. The waitress brought their drinks. The iced coffee was in a tall glass, and she spilled some of it as she put it down. Reacher's was in an insulated plastic carafe, and she shoved an empty china mug across the table next to it. She left the check face down halfway between the two drinks, and walked away without saying anything at all.
The Yahoo! Answers community also speculate as to its purpose, with some users rationalizing that:
...the check is presented in the middle of a table and face down (so) whoever has chosen to pay can do so discreetly without the other party knowing how much the meal cost.
...so the person paying can keep how much they paid private from others at the table.
...if one person is paying, it's polite not to show eveyone else at the table the cost of the meal.
So, for matters of etiquette, custom, or otherwise, checks are often given face down. This would necessitate the person who is footing the bill to "flip" the check/bill over to see how much is owed. A combination of the forgotten origins of this idiom, coupled with this social norm, may have driven many to question whether the term foot is correct, leading to the usage of an as-yet etymologically recognized saying "flip" the bill. Michael Quinion mentions an "odd feeling" that may hint as to why some people would question it enough to use the word flip instead:
It is an odd expression, isn’t it? It’s the kind of idiomatic phrase that we may use regularly without any feeling that it’s in the least odd, until somebody such as yourself asks about it.
One may argue that this is simply misinterpreting the idiom, but it has steady use. A Herald Tribune article  written in 2005 has the title NFL, Bucs aren't offering to flip the bill, regarding who would ultimately pay for legal fees in a court dispute.
More personal usage can be found in an article written for a Poker website and various posts to message boards and blogs:
The gaming friendly jurisdiction of Gibraltar is reviewing possible actions to have this bill ratified according to Gambling911.com sources and PartyGaming could potentially help flip the bill.
My question is: if we are competing for the hearts and minds, then are we going to have to flip the bill for the average Arab to have full access just so pro western ways aren't completely censored from the masses? 
That's why we must make sure that the health care plan that succeeds this year is a more comprehensive plan that DOES NOT FLIP THE BILL ON WORKING PEOPLE while corporations get off SCOTT FREE! 
In fact, a quick search of the exact phrase in a search engine shows how often people use this variant to mean exactly the original idiom's modern-day interpretation as described by Michael Quinion. And for some more familiar with flipping a bill after dining out versus footing the total on an accounting ledger, this might easily seem to be the accurate wording. Ironically enough, another Yahoo! Answers question discussing a matter of etiquette, specifically which party should pay for a dinner, received the following response:
The person who extended the invitation should flip the bill.
As of this writing, flip the bill has not officially been recognized as a variant of foot the bill, or even as an idiom in its own right.