Whether you can cash or deposit a torn check depends on the institution cashing the check and the check's exact condition. You can visit your bank to ask if they're able to process a damaged check and, if necessary, ask whoever wrote the check to issue a new one.
Evaluating a Torn Check
Different banks have different policies about torn and otherwise damaged checks. If you have one that's damaged, you can take it to the bank and ask whether they'll be able to process it. Some banks have stricter policies for mobile deposits, whereby you deposit a check digitally by snapping a picture of it with your phone, so it may be worth taking a torn check to the bank even if you normally use mobile deposit.
If the printed account and routing numbers on the check have been damaged, the bank is less likely to be able to process it. That's because machines generally process those parts and a tear can interfere with a machine, making it more of a hassle for the bank to process.
Some banks may offer to handle a torn check for an added fee. Of course, if you have accounts with multiple banks, it may be worth checking with more than one to find a bank that can handle a particular damaged check.
Replacing a Torn Check
If you find that your bank is unwilling to deal with a particular torn or otherwise damaged check, your best bet may be asking the person or company who issued it to replace it. Some institutions that issue a lot of checks have standard policies for replacing lost or damaged checks.
If an individual wrote you a check that became torn, he will likely want to make sure that it can't be cashed in the future. If you return it to whoever issued it, you can both avoid the costs of asking the bank to formally stop payment on it, which can be substantial, and the person who issued it can destroy it himself.
Replacing a Torn Money Order
Money orders are similar to checks, except that they are prepaid rather than linked to money in a particular bank account. Some institutions will also refuse to deposit or cash them if they're damaged.
Unlike a check, though, someone who paid with a money order can't simply destroy it and issue you a new one, since it's prepaid and that person will be out the money. If a money order is damaged, ask the person who issued it to take it back to the institution that originally issued it, like the U.S. Postal Service or Western Union, and ask to have it reissued.
In some cases, there may be a fee for replacing a money order.