If I drop to 0 points in a tier, how many games I have to lose to drop to a lower tier?
It seems to be about three in a row, though also note that right after you are promoted, you are given a grace period where you are not able to be demoted again.
Also if you make it through the five game promotion series to the next tier (tier as in silver, gold, etc.) you cannot drop back a tier. Meaning once you get to Silver V or Gold V, you cannot drop to Bronze I or Silver I(by losing games) respectively.
It is however possible to drop a tier after you have not played a ranked game for 28 days, at which time you will begin to lose League Points every 7 days after. The amount lost is different depending on your current tier, but if you have less than the amount that would be deducted from your current LP it is possible to lose your Silver, Gold, Platinum, Diamond and Challenger tier ranking. It is impossible to be dropped below Bronze V as this is the lowest rank currently.