If your app is published under US law, then the DMCA would apply, just as if it was a web site. The DMCA doesn't say anything about what particular technology the distributor is using.
TO be protected by by the DMCA's "safe harbor" provision, you will need to include a notice in your app that you accept takedowns, and provide an address or method by which they can be sent, and an agent who will receive them. (You can be your own agent if you choose.)
When and if you recieve a take down notice, you must check if it is valid in form. According to this Wikipedia article, a takedown notice must include:
(i) A physical or electronic signature of a person authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.
(ii) Identification of the copyrighted work claimed to have been infringed, or, if multiple copyrighted works at a single online site are covered by a single notification, a representative list of such works at that site.
(iii) Identification of the material that is claimed to be infringing or to be the subject of infringing activity and that is to be removed or access to which is to be disabled, and information reasonably sufficient to permit the service provider to locate the material.
(iv) Information reasonably sufficient to permit the service provider to contact the complaining party, such as an address, telephone number, and, if available, an electronic mail address at which the complaining party may be contacted.
(v) A statement that the complaining party has a good faith belief that use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.
(vi) A statement that the information in the notification is accurate, and under penalty of perjury, that the complaining party is authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.
(See the actual text of the relevant section of the law.)
If you receive (through your designated agent) a valid takedown notice, you must promptly remove the content and notify the poster (or you can instruct to poster to remove it, but you must do so yourself if the poster does not). If the poster then files a valid counter notice (see the linked sources above) with your agent, you must notify the sender of the original notice, and if the sender does not notify you of a copyright suit filed within 10-14 days, you must restore the content.
Provided that these rules are complied with, the host gets a 'safe harbor" and cannot be sued for copyright infringement, nor for the act of taking down the content. I believe that the agent must be registered with the US copyright office.
The courts have not ruled on just how quickly an ISP or other host must react to the takedown notice. It must be "expeditious".
Moreover, Under the DMCA (i) 1) (a) The host must have, post, and enforce a policy denying access to repeat infringers, or lose safe harbor protection. The text of the provision is:
(i) Conditions for Eligibility.
-(1)Accommodation of technology.
—The limitations on liability established by this section shall apply to a service provider only if the service provider—
--(A) has adopted and reasonably implemented, and informs subscribers and account holders of the service provider’s system or network of, a policy that provides for the termination in appropriate circumstances of subscribers and account holders of the service provider’s system or network who are repeat infringers;