If I understand correctly, you are asking whether a copyright holder's failure to submit a DMCA "takedown" notice in accordance with 17 U.S.C. § 512 prevents that copyright holder from enforcing their copyright. The answer is no, for multiple reasons.
DMCA is an optional process
First, a common misconception seems to be that the DMCA creates a mandatory process for anyone. It doesn't. The DMCA takedown process doesn't impose an obligation upon anyone: copyright holders don't have to submit DMCA takedowns, and service providers don't have to honor takedown notices. In fact, all 17 U.S.C. § 512 does is it provides a "safe harbor" for service providers who do follow the takedown procedures. In other words, with limited exceptions, DMCA says that internet service providers can't be sued for hosting content that infringes copyright unless they fail to obey a DMCA takedown first. But if the copyright holder isn't looking to sue the service provider, then filing a DMCA takedown is not a prerequisite.
Not enforcing copyright now doesn't prevent future enforcement
The broader question that you're getting at is also whether it's possible to lose copyright by failing to take enforcement actions in general, not only as authorized by the DMCA. This is a pretty common question because it turns out there's something sort of like this when it comes to trademarks, which are a different form of intellectual property. It's possible to lose trademarks for a variety of reasons: under 15 U.S.C. § 1127, trademarks are abandoned if they cease to be used or if they become genericized, for instance, so failure to enforce the trademark in one instance might cause problems enforcing it later. But copyrights and trademarks are very different, and copyrights are not abandoned in this way. It is actually fairly difficult to lose copyright protection: in countries that have adopted the Berne Convention (essentially every country), copyright exists from the moment a work is created, and though it is possible to abandon copyright, it must be done intentionally and the intent to do so "must be manifested by some overt act indicative of a purpose to surrender the rights and allow the public to copy". Hampton v. Paramount Pictures Corporation, 279 F. 2d 100 (9th Cir. 1960) (citing National Comics Publications v. Fawcett Publications, 191 F. 2d 594 (2nd Cir. 1951)).
As to your question about the rules of other jurisdictions, the Berne Convention's mandate that "enjoyment and the exercise of these [copyright] rights shall not be subject to any formality" (Article 5) would suggest that other countries also do not impose additional requirements for the full exercise of rights under copyright law, but the rules of any given jurisdiction vary and it would be impractical to discuss the steps required to enforce copyright in every jurisdiction in the world.
