First of all, it's important to keep in mind that Russia's total money supply is really small compared with America's. A quick glance at last year's data shows that Russia's money supply was only about 3% as large as America's. And that was calculated at official exchange rates--an iffy way of calculating things. Despite the immense size of Russia, its economy is pretty small.
So, use of the dollar in Russia might well have some effects on U.S. open market operations, but no matter which theory you buy, the effects would probably be pretty small.
There are different ways that a country can "use" the dollar. Certain countries, such as Liberia and Panama, have pretty much used the dollar in place of a national currency. U.S. currency circulates from hand to hand in these countries. If the Russians actually began using U.S. paper money (and plenty is already there and elsewhere), then this would in theory affect the relationship between the quantity of U.S. money and the level of U.S. prices. The quantity of U.S. currency in circulation would increase, but this increase would not place direct pressure on U.S. prices to rise (because these dollars would be held outside the U.S. by foreigners). There are some indirect paths back to U.S. prices, but I won't get into them.
Another possibility would be for the Russians to establish a currency board (like Argentina's) where the Central Bank's reserves would consist of deposits in the U.S. banking system, and rubles would be restricted to some multiple of those deposits (perhaps one-for-one). These Russian deposits would affect the U.S. monetary base. Therefore, the Fed would want to track these numbers and, potentially, to incorporate them into the conduct of monetary policy.
Since U.S. monetary policy is currently based around interest rates and not around the monetary base, though, it's unlikely that Russia's monetary situation would affect our monetary policy a great deal. No matter how much Russian monetary aggregates bounce up and down, they're not likely to have much effect on U.S. interest rates. But, in theory they could have some effects, so they would be watched.
One final note: For Russia to go to a dollar-based economy, they'd need to have some means of buying the dollars in the first place. They certainly couldn't buy any more dollars than the dollar equivalent of their current money supply. And in reality, they could only buy a fraction of that amount.
Note: Provided by EquilibriaChat, courtesy of the Federal Reserve Bank of Richmond. Please read their disclaimer.