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.
For cooking in with the pasta, consider red or yellow bell peppers, carrots, cauliflower or other vegetables that hold their color well when heated. Also, you can help this problem a lot with good, flavorful, beautiful garnishes. Even a simple shower of minced parsley goes a long way. Lemon or orange zest is nice too.
Check out this peppery red-wine capellini from my blog, you'll see how the quick, fresh garnish of cherry tomatoes and parsley turns a brown pasta bright (and maybe learn a new way to cook pasta too).
Best Answer
Based on comments, the likely culprit is moisture cooking out of the mushrooms and into your other ingredients. Mushrooms contain a surprising amount of liquid, and when cooking them you'll see that they shrink down significantly due to moisture loss. If you're adding them to other ingredients, some of the resulting liquid is hanging around in the pan long enough to discolor other, lightly colored items like eggs or tofu.
What I would suggest is sauteeing your mushrooms by themselves for at least a couple minutes, to release their moisture prior to adding everything else. Here's a good step-by-step guide. Be sure you salt them to draw out moisture, and make sure that you don't crowd the pan (see this question) so that the moisture can cook off effectively. Both eggs and tofu should cook fairly quickly, so you can add them after the first couple minutes and the mushrooms can continue to brown and cook afterwards. If you have other ingredients that need to cook for an extended amount of time, you may want to cook your mushrooms separately if the color bothers you. If you're using a separate pan, this won't really add additional time, but you will have extra cleaning to do.
Mushrooms are funky that way - common sense might indicate that they be cooked like vegetables, but there's a reason why Alton Brown suggests treating them like meat instead.