Yes. In fact drinking it will be better for your teeth than any bottled water because of the flouride that is added. However, if you brush and you just don't want to spend your life carrying those giant bottles around-- invest in a water jug with a carbon filter in it as 'Hana VK' suggests. They're available from most big stores, are inexpensive, and if you leave it around visibly in your apartment it may encourage you to drink more. And here's the most important thing: For best health outcomes, until your pee is clear.
There is no immediate consequence to drinking the tap water; however if in Moscow, a filter or a couple of bottles of water certainly won't be your biggest investment.
Conflicting advice about water quality in Moscow is rife. On the one hand, you have authorities at water companies vehemently stating that the water is safe, sanitized and perfect for imbibing. On the other, restaurants only serve bottled water, most Russians drink bottled water and most guidebooks scream "don't drink the water!" So which is it? While there is proof that the St. Petersburg water supply has contained giardia, a nasty intestinal parasite, there is no conclusive link to Moscow. The yellowish tinge to your bathwater is apparently down to additional groundwater run-off in the spring when the snow melts, and while unsightly, it is not necessarily harmful. That said, pre-Soviet era pipes remain in use and there do seem to be raised nitrate levels in Moscow water. That isn't terribly bad for your health unless you're pregnant — then it's not advisable. With a basic supermarket water filter costing less than 500 rubles, you're better safe than sorry. But for brushing your teeth, washing your fruit and summertime water fights, you'll be absolutely fine.