The unexpected behavior appears to result from the fact that you are using headphones, which the monitor does not technically support.
It seems odd that you would receive mixed results. Especially when the the monitor has been confirmed to support consoles. That said, I am finding confirmations for the PlayStation 4, not a PlayStation 3. One might expect the signal to be more reliable in the newer generation of console.
That said, I found a source of confirmation on the official Dell forum, that may shed some light:
Posted by DELL-Chris M on 2 Sep 2014 6:41 AM
Thanks for confirming this. The Audio-Line out should be able to
transmit audio from HDMI, mDP, DP sources. There is one notation in
the User's Guide,
Audio-Line out = Connect your speakers
Headphone usage is not supported for the audio line out connector
You may need to make sure your audio source sound is
set to the video output you are using.
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So why doesn't it work?
The cause, at this point, is mostly speculation. Ultimately, you can not expect an unsupported device to behave as it was intended. That said, I can account for the sound output of television devices and computer devices to be very different. I have seen speakers perform well for a computer, only to distort to the point of usability, on a television.
I would speculate that your computer is providing a strong enough audio signal to pass along into your headphones, but the signal from your console is simply to weak. This is supported by the documented requirement of providing speakers; A device that actually adds power/gain to the audio signal. Think of it this way: The audio is there, but it is very quiet. Your headphones do not have the power, and likely the capability, to raise the volume. Speakers often do.
How do I fix this?
Plug a pair of decent speakers into your monitor, instead. You can get some reliable stereo Logitech speakers fairly cheap1. Plug the speakers in to your monitor, and raise the volume up, to adjust with the console.
But what if I really want to use my gaming headset?
Assuming the speakers solution works, all you have to do is 'piggyback' off the speakers. Ensure your speakers have an earphone out connection (very common, but ensure the model you purchase has the feature, just in case), and plug your headset into the speakers. Assuming the speakers emit volume, the headphone literally becomes a matter of plug in and play.
1Although, not necessarily from the manufacturer, themselves. Shop around. I have seen some of these speakers for a quarter of the value, in store.