I think if the pitcher is going to accomplish a no-hitter, it's his responsibility to accomplish it. If you're a hitter with a hit streak going, why dog it at all? Go for the hit; if the pitcher deserves the no-no, he'll get you out anyway. A difference in this situation would be if the pitcher going for the no-no in the 9th walks that player with hit streak; ouch.
Now if it's a perfect game situation, and I was the one with the 25 game hit streak, I would probably tone it down and let the guy get the perfecto; those are once in a lifetime opportunity and will get you quite a bit of prestige. A hit streak barely means anything unless you get close to 50 games.