You may study your appointment letter and check if you have signed on Non disclosure, Non Compete clause. Company might have added that employee shall have to sign e.g. non disclosure clause at the time of termination of employment contract.
Depending upon the language you may prepare your defense. You may not put anything by a statement on record while working with new employer which can be used against you by old employer as revealing their trade secret, sensitive information.
Relieving letter is issued post all settlements. Settlements due at the end of employee are, to complete tasks on hand, return company property/handing over the charge (under proper acknowledgment), advance/loan/imprested cash ( which company can adjust in FNF statement also if amount is sufficient to adjust), any other item on record.
Relieving letter is more of a trend, practice which states that employee does not owe any thing to the previous employer.
Company can sit over the relieving letter. Acknowledgment of notice/resignation/company property, acceptance of resignation, FNF statement, settlement and payment of FNF dues, last salary slip, can serve as good as relieving letter.
It shall be appropriate to tender reasonable notice and avoid abrupt termination. Remain amiable, vigilant, and do not leave any room for company to charge you on any count.
Some companies press employee to sign a non disclosure agreement at the time of separation and apply coercion like if employee does not sign relieving letter shall not be issued.
Employee should inform the next employer that current employer (preferably in writing under acknowledgment) is unwilling to issue relieving letter and list of documents employee can supply. New Employer can accommodate the new employee without relieving letter. Be smart.