.. the subject line says it all really.

Maybe it's the cold weather, but quite often I press the button on the dash, hear a nice big 'click' but the glovebox door refuses to drop open. It sticks and I have to prise it open (gently) with my fingertips.

And yet sometimes it's ok.

Is this a common / fixable problem or is there a trip to my Audi dealer in my near future?

 

It is no wonder why the population gets more and more sedentary, and why they call these cars "luxurious". They just want to prevent elderly from getting tendonitis in their fingers. It took them only ten years (your car is a 2005, right?), but finally their ergonomics department got the design right. From a mere handle they went to a button on the dash board. That's so convenient, especially if the driver wants to open the glove box while s/he drives...
Ok, ok... I had my little amusement

Leaving the joke aside, it has to be the locking mechanism that is sticking in a closed position.



Before rushing to the dealer, it may be probably a good idea to try "detoxifying" it yourself. Maybe the easy clean-up job will cure the problem. I don't know how the glove box looks like in your year model, but probably the latch mechanism is similar.
I can take mine out via two Pillips head screws, as it can be seen in the photo.


The latch appears to be plastic...Probably a little dab of lithium grease may solve the problem.
Good luck.

 

Thanks Chefro, I'll take a look at that!