Do apps placed in the background consume resources? I've heard that leaving open apps on an iPhone, affects the duration of the battery's charge. Is this true and documented, or some of an urban legend? And does it affects other apps' performance?
Third parties Apps which are placed on the background won't actively consume resources unless they provide either Location Services, Audio playback or VoIP. As stated on Apple's iOS App Programming Guide:

In iOS, only specific app types are allowed to run in the background:

Apps that play audible content to the user while in the background, such as a music player app.
Apps that keep users informed of their location at all times, such as a navigation app.
Apps that support Voice over Internet Protocol (VoIP).
Newsstand apps that need to download and process new content.
Apps that receive regular updates from external accessories.


There are, tough, some Apple services which will, too, run on the background and consume resources, such as Photo Stream (upload / download), App Store downloads or even push notifications.
In the case you've got a lot of backgrounded Apps, and the currently open one is running out of resources, iOS will, automatically, close some Apps freeing memory. Extracted from the same Guide:

When a low-memory condition occurs, the system may purge suspended apps without notice to make more space for the foreground app.