Is there any way to tell iOS mail app to put emails from specific senders in a special folder automatically?
If you’re using an iCloud mail account then you can go to http://icloud.com/#mail
In the top right of the screen there’s a preferences icon — looks like a cog. If you click on that the preferences pop-over opens, click on Rules and from there you can configure some basic rules to put messages into specified folders.
If you make them a VIP and use the VIP folder - YES.
If you don't use VIP, no - not in the mobile Mail app. You'll want to look into Gmail and the gmail app for that sort of server side filtering.
Largely, the answer is "No". iOS default mail doesn't have some of the things that OS X Mail has such as rules. The good part is there are a few work-arounds.
Depending on your needs, there are VIP mailboxes in iOS 6. These can be configured by tapping Mailboxes (back arrow) in the Mail app, and then tapping the arrow next to VIPs. There are several problems with this method, however. The biggest being, it puts those emails into the VIP folder AND the inbox. The other problem is it is just one "folder".
For many reasons, VIPs may be undesirable you, and the only other way to manage that would be to manually move them. Tap the Edit button above and on the right of the email list. Tap the emails you would like to move, and then tap Move, then select the folder to move them to.
Side note: the only other way to do this would be through a service like Gmail, or using the OSX mail app.
Just another side note: Rules can be used indirectly. If you use preference rules to sort email with the Mail.app on an Apple computer, the listing of emails in your inbox can be updated and sorted into folders on your iPhone, by opening the Mail.app on your phone using the icon that appears on your resting screen of the phone. This syncs the email on your phone with what you have done on the computer.
I have a method that works using archive folders within an IMAP or Exchange email accounts. Manually create your archive folders somewhere within your email folders list. The archive must be stored online, not locally. I use Outlook to create my folders manually, but i think it would work without Outlook as well like on your web access. Once the folders are created and you've previously moved messages to that folder, your iPhone remembers this. So in the future once you've previously moved messages to these folders, new emails you get from that sender sender can be moved there very easily by using the move message feature inside mail. A pop-up will appear with 2 options; the previously used folder to quickly move the message there, or an option to select a new folder. This usually works for different types of emails from the same sender as well, say you have a subfolder for "Orders" and "Shipping Notifications", it will remember which folder that email normally belongs. It's not automatic, and its not perfect, but very easy to do and pretty flawless.
I have a solution that may not be for everyone as it involves Outlook on a permanently on PC. I use IMAP on my phone, the mail account in Outlook is always on and many rules in place, since the PC syncs every minute or so the rules are applied by Outlook. I get the occasional one that lands on the phone prior to the PC so does not get action-ed, apart from that it works well.