Probably none; microchips encased in plastic are pretty tough, and freezing temperatures ought to be within the limits of conditions the card is designed to withstand (e.g. it would get pretty cold if it’s on side of the wallet away from your body heat during cold winter weather).
I would definitely keep the card. Credit reports look at age of accounts and at peak utilization, so having an account in your history that you charged up and then paid off is generally a good thing. If there’s an annual fee, talk to the company about options to avoid that - often times, they can change the type of the card or will agree to waive it.
Another good reason to have a credit card (on you, not just in the freezer) is in case of emergency. For example, if your car breaks down in the middle of nowhere, or an airline cancellation strands you at some strange airport overnight. Obviously, you need to have the self-control to identify what is a genuine emergency worthy of using the card and avoid using it on frivolous things, but having the card there when you need it can be a real life saver.
My checking account has a credit card that is linked to it as overdraft protection. I never activated the physical card (you could destroy it), but the account is still open and it lets me use the debit card as my only credit card without worrying about running out of money in an emergency. You might see if you bank offers something like that.
No, I think I’ll have to just keep the stupid account for the original card. But there WILL be a Credit Card Cutting-Up Party. I really like the idea of the Visa-shaped cake. (Ponders the idea of vanilla with chocolate icing…)

Unless there’s an annual fee, keep the account. If there is an annual fee, get one with a different bank or credit union, where there is no fee.
Average account age helps your credit rating. Something to do with lower average age means more credit applied-for recently. I’ve still got an account I opened in college, because it really jacks up my average account age, even though the account is more of an annoyance than anything else.
Your credit rating may not be an immediate issue, but if you’re looking to apply for a mortgage somewhere down the line, having a long, well-established credit history will help.
This is what I would do, unless you really, really lack the self-discipline to do it. I had some credit card problems about 7 years back, culminating in one of the accounts being 4 months overdue, the amount overdue being well over the limit, and the credit card hounding me. We worked out a payment plan (lowered interest rate), and I managed to whack that debt off the card in about a year and a half.
Now, I didn’t close that card. In fact, I use my card for practically all my purchases for many reasons discussed above, plus because I get 1% cash back on the purchases. I have $500 in insurance on items stolen/broken/lost with the first 90 days of purchase. I have insurance on rentals. I have some kind of travel assistance. Etc. I just make sure to log in every couple of weeks and pay it off. I opened a few more accounts so my debt-to-credit ratio is low. I accepted every credit limit raise they offered me, and even requested a few, because, hell, I’m not going to use it and it looks better to creditors. If I have $500 in limits on my revolving credit lines and I’ve charged up $400, I’m at 80%. If I have $20,000 in available revolving credit, I’m only at 2%. This helps your FICO score, should this be important for you (like farther down the line when you want to get a mortgage). My Experian and Equifax scores are now over 800.
Now, I don’t know exactly what your relationship with debt and credit cards is. But if this was just a temporary road bump in your financial situation and you can avoid the temptation to just run up your cards at every chance, I would keep the cards. Credit cards are not inherently evil and are better than cash for a lot of reasons already stated–provided you pay them off regularly, do not use them for cash advances, and don’t run them up.
Be aware that they may close the account if it is not being used, especially these days. I had one closed without any notification because I had not used it in a year or two. My credit report now shows it as ‘Closed at customer request’. (I had failed to activate it, so that might have been the problem. So, if you shred it, when you get a new one - activate before shredding it again).

As others have said, the protection is better on a credit card than a debit card, even if it is a visa debit card. If somebody racks up your credit card, you just have a large balance until it gets cleared up. If somebody charges on your bank card, you have no money until it gets cleared up.
I think I understand your intent, but your words are incorrect.
Credit Card Loss or Fraudulent Charges (FCBA). Your maximum liability under federal law for unauthorized use of your credit card is $50. If you report the loss before your credit cards are used, the FCBA says the card issuer cannot hold you responsible for any unauthorized charges. If a thief uses your cards before you report them missing, the most you will owe for unauthorized charges is $50 per card. Also, if the loss involves your credit card number, but not the card itself, you have no liability for unauthorized use.
Lost or Stolen Credit, ATM, and Debit Cards | Consumer Advice
No matter what happens, the total loss to you cannot exceed $50.00.
On the other hand …
**ATM or Debit Card Loss or Fraudulent Transfers (EFTA). **Your liability under federal law for unauthorized use of your ATM or debit card depends on how quickly you report the loss. If you report an ATM or debit card missing before it’s used without your permission, the EFTA says the card issuer cannot hold you responsible for any unauthorized transfers. If unauthorized use occurs before you report it, your liability under federal law depends on how quickly you report the loss.
Lost or Stolen Credit, ATM, and Debit Cards | Consumer Advice
ATM or Debit card losses are based on time:
For example, if you report the loss within two business days after you realize your card is missing, you will not be responsible for more than $50 for unauthorized use. However, if you don’t report the loss within two business days after you discover the loss, you could lose up to $500 because of an unauthorized transfer. You also risk unlimited loss if you fail to report an unauthorized transfer within 60 days after your bank statement containing unauthorized use is mailed to you.
If you don’t follow the reporting procedures in time, the account tied to the ATM/debit card can be cleaned out.
The best strategy is to keep one, or even two credit cards, providing they have no non-ballance related charges whatsoever. Each year, use each one once, and pay the entire ballance immediatly upon reciept of the bill.
A gas company card is generally associated with a rebate, but read any fine print very closely before signing up for one. Your bank will probably issue you a credit card if you apply for one. (My bank allows me to assign the card as an automatic choice in the event of an overdraft, and I do so, although I avoid writing checks as if it was available.)
Will power is better than preventing temptation.
Tris
Your credit rating may not be an immediate issue, but if you’re looking to apply for a mortgage somewhere down the line, having a long, well-established credit history will help.
It’s good. That one credit card got run up out of… well, sheer dumbness is the only way to put it. As for the mortgage thing, well, let’s just say that I’m starting to think I’d be better off paying cash. And believe me, considering where I am, what the foreclosure trend is around here, and the staggering drop in home prices in this area, the day will come sooner rather than later. (But not right now-- they will still go further down. I would bet anything on that.) Never again will I take out a loan for anything, never, never, never; yes, I know, .000001% interest rates, la la la, I don’t care how good the interest rates look, you’re still buying on time and it’s not worth it.
(I’m still set on the CCCUP (credit-card-cutting-up party, even though I don’t think I’ll get rid of the account. I just like the idea of the Visa cake too much to give it up.)

Never again will I take out a loan for anything, never, never, never; yes, I know, .000001% interest rates, la la la, I don’t care how good the interest rates look, you’re still buying on time and it’s not worth it.
This is not necessarily true. Hell, if you can get a .000001% interest rate it would be utterly stupid not to take that loan, unless you think the currency is going to deflate.
But obviously, you’re exaggerating there. Still, there are sound financial reasons for taking a low-interest loan rather than paying it all off right away. If you feel investing the money you would have spent on the loan will outpace the loan interest, in the long run, the loan may be the more financially prudent option. Of course, there is some risk involved, but if you follow historical returns on the S&P 500, if I had $250,000 in cash for a house I would take a 30-year-loan at just under 5% and dump the rest in to an S&P index fund (with an 11% annual return from 1950-2010 or 7% inflation adjusted) vs. paying the house outright. Depends on your personality and financial outlook.
Keep it, keep it in your wallet. Could come in handy, won’t hurt anything.