It's important to note that wash or no wash, mushrooms themselves already have a high water content so their ability to absorb more water is limited. Estimates I've found put the range between 75-90% water, depending on the type of mushroom. Cooks Illustrated, for example, puts it at 80%.
Sources such as Cooks Illustrated, Alton Brown, Harold McGee, Robert Wolke, and Kenji Lopez Alt have tested washing mushrooms by weighing them pre- and post-wash, and all found that mushrooms absorb a minimal amount of water that does not significantly affect cooking time. Both McGee and Wolke soaked the mushroom for five minutes - so much longer and with more direct contact than than a normal simple rinsing. Lopez Alt's testing, for example, found that the mushrooms absorbed only about 2% of their total weight which translated to an extra 15 to 30 seconds of cooking time. After testing, these sources all encourage washing mushrooms.
Several of the sources also observed that most of the additional weight is on the surface of the mushroom, which can either be patted dry with a towel or paper towel, or removed with a salad spinner, as other previous answers here have noted.
Cooks Illustrated suggests only washing whole mushrooms, as when cut they can become more absorbent. They also suggest that if you are serving the mushrooms raw, use a tooth brush to brush them clean rather than rinsing because rising can cause discoloration.
Washing should be only done just before cooking. Washing and then storing mushrooms can shorten their shelf life and they can become slimy on the surface.
Wolke notes that if your mushrooms are steaming rather than browning, it's more likely that your pan is too crowded rather than a function of having washed them.
Cooks Illustrated describes their process here, McGee in The Curious Cook, Wolke in What Einstein Told His Cook, Lopez Alt in The Food Lab (and also Serious Eats) and the best I could find for Alton Brown were transcripts from the Good Eats Fan Page. Both the the Cooks Illustrated and Serious Eats links also have more general tips on mushroom storage and prep.
Soak them for a while in warm water, and you'll be able to use them, yes.
However It is equally important to know that you should reserve the liquid for its essential mushroomness, and yet also that said reserved liquid should be run through a coffee filter to remove grit.
Best Answer
You cannot avoid the foam, it's going to form but you can keep it from boiling over. A foamy pot boils over because the foam traps steam, acting like a lid sitting right on top of the water. Escaping steam takes heat away from the pot, preventing it causes the water to super heat and boil over. You can prevent it by simply stirring, and floating a wooden spoon on the top (actually works).
However, I would prevent this by reducing heat. You don't need to boil your food, a simmer will work just fine and is more gentle on your ingredients, less heat means even with foam it won't boil over. So turn it down and solve your problem.
As for why the foam forms with mushrooms I don't know. With pasta and rice it's the starches on the surface which come off and expand, but mushrooms aren't starchy. I suspect it's other particles on the outside of the mushrooms, or perhaps the spores under the cap. To reduce foaming you could try rinsing the shrooms, that's usually a no-no but as you're cooking them in water it's no big deal. You could saute them separately if you have the time, you'll get a different flavor which you may prefer, although you wouldn't get it that way in a traditional Japanese curry.
Here's an article about foaming in pasta water which may be worth a read.