When using Sonar X3e --

I want to hear the sound of an echoed audio track [with VST effects] when I have selected a completely different application that is not Sonar (e.g. a web browser).

Is this possible?

I have sonar set to share audio drivers with other applications, but the audio of echoed tracks still cuts out when I am in another application's window.

Sure. I'm guessing you are trying to use a Synth which has a different input echo setting (automatic) which requires the track to be selected to input echo. To work around this, just click the input echo button a couple of times and it will remain on (without the A). This means it will still input echo even when the application is not selected.

Thanks gswitz!

I am actually doing something more complex -- I am using Sonar much like Ableton would usually be used, to process live audio from a microphone with VSTs -- audio which is then being fed out in real-time through my sound interface (in this case, my laptop's RME Babyface). 

However, I *also* want to use the same laptop to output a web-browser-based sound effect and animation so I can HDMI-capture the laptop screen on my main broadcast machine with the sound output of both Sonar AND the web browser. 

Unfortunately, whenever I click an application window that is not Sonar, Sonar's echoed track audio disappears -- even when ASIO drivers are shared AND the echo button is on. 

Oops... just read more carefully.
 
Ok, set your outputs for Sonar to be idk, 1+2 and the outputs from the browser to 3+4 and bring them together in TotalMix. Or, you could route Sonar to the SPIDF Out and the browser to the ADAT out and in TotalMix, merge the two together and route to an analog out.
 
BTW, you can record using DigiCheck (which comes with your RME).
 
http://www.rme-audio.de/en_downloads.php?page=content/downloads/en_downloads_driver&subpage=content/downloads/en_downloads_digicheck
 
You can record the output of Sonar and the output of the web browser independently using DigiCheck (along with all the inputs into your Babyface).

You are using Asio, right?

tayzonday
When using Sonar X3e --
]
I have sonar set to share audio drivers with other applications, but the audio of echoed tracks still cuts out when I am in another application's window.

You *dont* want to do this. This is telling SONAR to release the driver when it loses focus which is why the audio cuts out.

Noel, I'm totally puzzled.
 
I created a new project and added TH2 to an audio track and started playing. All was well.
I switched to Media Monkey, kept playing. All was well.
I switched to Chrome, all was well.
I switched to DigiCheck, all was well.
 
I'm not having any trouble doing this. I don't understand why he shouldn't do this.
 
If you really couldn't input echo outside of Sonar, how could you flip over to TotalMix and tweak the output routing while playing your guitar? You couldn't, and I do. It works fine. I'm totally puzzled.

btw, I do not have 'Share Drivers with other programs' checked.
***
I think I just came to understand Noel.
He is saying, 'Don't check share drivers, because that tells Sonar to let go!'
 
I just tried checking the 'share drivers' button and it causes audio to stop when you change programs.
 
Now, that doesn't mean that other programs can't route audio. They can. They just can't change the sound interface settings (like sample rate).

Noel Borthwick [Cakewalk]

tayzonday
When using Sonar X3e --
]
I have sonar set to share audio drivers with other applications, but the audio of echoed tracks still cuts out when I am in another application's window.

You *dont* want to do this. This is telling SONAR to release the driver when it loses focus which is why the audio cuts out.

THIS was the problem! Thanks so much! Now Sonar outputs no matter what window I'm in. I guess I misunderstood how that checkbox works (that "sharing" actually means "giving up and handing-over" . . . not "use together with everything else." )  

I agree the terminology is a bit confusing - it dates back to the time when most interfaces were mutually exclusive and you couldn't use two apps in ASIO mode at one time with the device. Share just means "give up" use of the device when the focus changes to another app. 
There are some newer audio interfaces that are smart enough to support multiclient use by different apps using different sample rates. For those devices you can use them at different sample rates even with sharing disabled.
The RME devices work like that for example.