I agree with Perpend that "My bottle is full of beer" is the normal way to make a remark about that bottle and its condition. I'd only use "My bottle is filled with beer" if I wanted to remark on the process of filling the bottle or something similar. Such remarks are pretty rare in my speech.
'Full' tends to relate to how much is in it, where it is on the scale between full and empty: a bottle full of beer, a swimming pool full of water, a bucket full of sand are now at the maximum level, either because they've been filled up (the pool) or because none has been removed. 'Filled' places more emphasis on the action, so is typically used when the container does not normally contain the substance. A balloon filled with water is a balloon that has water in it (unusually for a balloon) - we're not contrasting it with a half-full water-filled balloon, but with an ordinary air-filled or deflated balloon. We'd probably prefer 'filled' for a beer bottle filled with tap water (perhaps to hold flowers): again, the interest is in what it has in it (what it's been filled with), not so much with the quantity used.