We live in an age with many trillion dollar elephants in the room and most of them have complexities that are not discussed in widely read publications. The conflation of quantitative easing and debt monetization by the media is one of them. The two activities have very similar effects in the short term, but their effects in the long run are quite different.
The Federal Reserve controls the money supply and Treasury handles government revenues, expenses and borrowing. They are distinct entities. If the Federal Reserve and the Treasury were actually merged, then the Fed could hand bricks of cash/bank account entries over to the Treasury Department in order to directly fund the budget deficit without the issuance of new debt. This would be an example of monetization of the debt. Another way that this could occur would be if the Treasured issued bonds to cover a deficit, the Federal Reserve printed money to purchase them, and then forgave the debt.
This is not how things are currently done. When the Fed engages in quantitative easing, it creates money out of thin air in order to buy financial assets such as treasury bonds, but it does not cancel the borrower's obligation to repay. So, does quantitative easing increase the money supply and cause inflation in the long run? That's really not so clear. Consider the following examples:
The Fed buys $100 billion worth of newly issued 2-year treasury bonds at par that that pay a 1% coupon rate. This results in $100B entering the money supply now, but assuming that the Treasury keeps making the payments, both the principal and the interest will leave the money supply and return to the black hole that is the Fed. This means that if the federal government doesn't default, then over the life of this trade, quantitative easing will add and subtract $100B in principal and subtract $2B in interest from the money supply, for a net reduction of $2B. If the Fed had instead purchased $100B worth of 30-yr bonds that yielded 4%, the federal government didn't default, and the Fed held this bond to maturity, then it would suck a net amount of $120B (4%/year*100B*30yrs) out of the money supply.
In the round of quantitative easing that the Federal Reserve began at the peak of our financial crisis, they bought a number of securities other than U.S. Treasury bonds. (Feel free to look at their current holdings here: http://www.federalreserve.gov/RELEASES/H41/Current/) These other purchases primarily consisted of bonds issued by Fannie Mae and Freddie Mac, which are now backed up by the federal government, and mortgage-backed securities issued by them as well as Ginnie Mae. Unlike bonds, mortgage securities have variable times to maturity. If you sell your house or attach and extra $100 to your mortgage payment each month, the loan gets paid off earlier than expected. If the borrower defaults and the house is sold, this also results in an early repayment of the loan, even though not all of the money owed is returned. The result of all of this behavior is that investors in mortgage loans typically expect to get most of their money back substantially sooner than three decades later. In the case of these government agency issued mortgage securities, the risk of default by homeowners is insured against by the agency (and hence the taxpayer), so the Federal Reserve doesn't really take any risk that they won't be repaid as long as the federal government doesn't default.
What's the summary of all of this? Through quantitative easing, the Fed temporarily increased the money supply with a built in mechanism to reduce it as the crisis abates by simply holding the bonds it purchased to maturity. They can also potentially speed up the process of reducing the money supply by selling the bonds back to the market, but most of the bonds that they bought are going to be largely repaid within a few years anyways. Is it possible that if the Fed leaves too much money in circulation as the recovery (HOPEFULLY) picks up steam that we'll see a rise in inflation? Yes, that's certainly a possibility, but Japan engaged in QE and they saw deflation happen, which shouldn't be surprising given that QE reduces the money supply in the long run, so the evidence would suggest that fears of runaway inflation are probably overblown.