Your outer layers shouldn't have to be washed unless they have come into contact with dirt - trains trailing on the ground, spills, etc. The inner layers are the ones that tend to need to be washed most often.
If you don't know what the content of the fabric is, it's going to be a crapshoot. If you know for sure that it's polyester, or a natural fiber like cotton or linen that HAS been pre-washed before construction, then you should be safe machine-washing the inner layers with a cold wash, possibly delicate cycle. But if you have no way of being able to tell what the fabric content is (whether it's natural or synthetic fibers) or whether the fabric was properly pre-washed before the commissioner started sewing, then you may need to err on the side of either hand-washing or dry-cleaning. Having strong contrast colors next to one another on the same garment (e.g. red and white) will mean it's probably not a good idea to just dump it in the washing machine, unless you know for absolutely sure that it was pre-washed or it's a poly that won't bleed or run - or shrink. Too much metal on the costume that isn't stainless is also a reason not to machine wash it, in case cheaper metals rust.
If you really can't find out what fiber content the fabric is (damask is a weave - is it cotton? Poly? Silk? A blend?), and don't have any way of snipping a tiny swatch out of the seam allowance to do a burn test, a delicate hand-wash or dryclean is probably the only thing you can do.
Founder and President, Madison Area Costuming Society, a chapter of the International Costumer's Guild
Strange Land Costuming - www.strangelandcostumes.com