This is a very bad thing.
Here's why in a super-tiny nutshell (more details and numbers here - also the source of the image above - and here): in a recession, there are fewer jobs than there are job-seekers. (We've been making improvments, but not enough yet.) This means that without unemployment, many many qualified workers - not deadbeats, mind you, but people who are simply victims of the numbers imbalance - will be pretty well and screwed.
These folks may cease to be able to pay mortgages or rents, cease to be able to buy significant amounts of food, and so forth. This money therefore does not go to the retailers and renters who depend on their customers and rentees. It's a domino effect of badness. The Congressional Budget Office recognizes how bad things would have been had we not had UI extensions previously.
Compare this to the tax cuts for the wealthy which the Republicans want to keep (while saying that we "can't afford" the UI extensions) - the wealthy tend to sit on their money, and that which they spend may well go to foreign countries via strange tax loopholes and business ventures or plain old vacations and importing. Regular joes who are unemployed spend much more locally.
I was unemployed for several months in 2009-early 2010 and again late this summer. The money I got from the government (some of which was unemployment extension from Congress), where did it go? Why, to groceries. Local dining. Gas. Rent. Utilities. Student loans. Household items. Local entertainment (largely theatre, which was then most likely spent by those artists on groceries rent etc.). Health insurance via COBRA. A car repair or two. I carefully balanced my budget to make sure my expenses did not excede my unemployment payments, so that I could keep my savings for luxuries or emergencies (I only ending up using about $200 of them). The point is - nearly every penny of the money the government sent to me went straight back into the economy. (By the way, most of these purchases did also incur sales tax, and the businesses I and those liked me helped support pay taxes, too.)
Now, the economy wouldn't work whatsoever if the government just took money from the working people, gave it to the non-working people, and expected that this money would then go back to the working people when it's spent on groceries. Someone has to generate the money in the first place.
The important thing here - the one super-important thing to remember is:
THERE ARE NOT ENOUGH JOBS FOR EVERYBODY.
During my unemployment, I applied to two jobs a week minimum. I took my time crafting those applications, customizing the cover letters, etc. It didn't matter. If there are dozens of qualified people applying for the same jobs, then dozens-minus-one of them are not going to get it. Period.
In this original post about my unemployment (which cooked up quite a little localized Internet firestorm, albeit with typical talking points) (and which I never realized was getting so many responses at the time - I should learn to watch for comments more closely), I mention how the prospect of going to work at Starbucks was not a pleasant one. Two things I neglect to point out in that post: one, Starbucks does not have infinite jobs to offer for unemployed knowledge workers/college graduates. Neither do Walmart or anyone else. During a recession, all the unemployed middle-managers and artmakers and decently-paid office peons (myself being in the latter two categories) can't just all suck it up and move over to Starbucks and Walmart. Especially because some of them couldn't feed their families or keep their houses on Starbucks wages (whereas unemployment gives them enough to do so). Nor can little part-time jobs always fill the gap, because some people - like me - can't afford health insurance without employer assistance or a significant paycheck ($350 of my unemployment checks went to my insurance, plus deductibles), and can't go without insurance for health reasons.
(The thrust of my original post was more complex than that, and, as a late response to those that misread it, was mainly critical of myself. The point of that post was to examine the psyche of someone in my position, not to argue for or against unemployment insurance. This post you're reading now is arguing for unemployment insurance.)
(Additional note - I am currently employed, thanks to our very own b.graham and ali d. Networking trumps both brute force resume-carpet-bombing and tactical cover-letter-strikes.)
Simply put: if you think that, during a recession, leaving the unemployed high and dry will either lead them to get off their lazy asses and get work, or at least give them their due punishment for being freeloading bums, you're wrong. (Even if, reading my previous post, you deem me to be in the lazy-freeloading category, there are millions who definitely are not.)
In a recession, preventing the bankruptcy and poverty of the qualified-but-unemployed prevents the recession from worsening and allows the economy to recover while maintaining quality of life. When the recession is over, that's when you get tough on the expense of unemployment and start looking for deadbeats.
In conclusion: letting these benefits expire does no good for anybody. The government only saves money in the immediate sense; in the long run, tax incomes are hurt, economies weakened, and everybody loses.
So the Republicans are either total hypocrites ($830 billion for tax cuts to the rich okay, $12.5 billion for UI extensions not okay!) - practically bald-faced in their commitment to robbing the poor to feed the rich - or they're stupid.
Or there's a third option (not incompatible with the other two): the Republicans may realize this is all a bad thing, and are doing it anyways, because the only thing that matters is making Obama lose in 2012. It's for the good of the country!