The bank on which the check is drawn should be able to honor it with no problems. However, if the recipient takes it to his bank, the processing machine might have problems reading the material at the bottom of the check.
The information at the bottom of the check must go through the bank's check-reading machine successfully for the recipient's bank to process a check from another bank, so if the tear interferes with that information, there may be a problem.
Taking the check to the bank on which the check is drawn is the best option. However, if that bank refuses to cash it because of extreme wear and tear or the check missing some of the data at the bottom, the recipient has to ask the sender for a new check.