My GLI has been plagued with worsening buzzing and vibration sounds from the dash area since it was new. Seems a little worse every day. I stumbled upon a post about unplugging the "Soundaktor" in a GTI and found several before and after videos on YouTube for the GTI. After doing a little Googling found that the GLI is also equipped with the electronic gizmo.
For those unaware, the Soundaktor consists of an electronic module that is wired to what looks like a small speaker that is bolted to the firewall area just under the cowling that is beneath the windshield wipers. This is the device that is responsible for generating the low rumbling sounds in the GLI, which I've never liked.
In 5-10 minutes, I removed the foam molding that secures the cowling, lifted the cowling, reached in to the speaker module and unplugged it. WHAT A DIFFERENCE! The GLI still has a sporty sound, but is much more natural and all of the low-pitched rumbling is gone, along with most of the dash vibrations.
If you're going to do this, be sure to pinch the bottom of the plug where the wire attaches to the speaker, and it will come right out. If you have to use force, you're doing something wrong.
I HIGHLY recommend giving this a try and think you'll like the new sound of your GLI. If I have time, I'll post a video somewhere.