You can try depositing a torn check, but if it's accepted or not depends on the bank. If the bank clears checks automatically, then the check may not be processed. Checks with small cuts or tears are generally accepted but ripped or Sellotaped ones may not be.
Cuts or tears that do not damage the code line at the bottom and that leave the signature or amount unharmed are almost always accepted by banks. Some banks may manually process your check, for which a small charge may be levied. For ripped checks, it is sensible to ask for a replacement. Check your bank's website for more details, or contact a customer care executive at a branch.