Doesn't matter whether it is rent control or minimum wage laws. Government set price controls distort markets negatively.
In this case, the total number of trips taken on Uber in Seattle is going to drop because a certain percentage of trips will now become uneconomical to those riders: they will cost too much to be worth it.
Two negative things will happen:
1. In aggregate drivers in Seattle will make less money due to fewer trips. Demand for Uber drops as the price increases. A simple thought experiment confirms this: lets say that Seattle demands a flat $25/tax per ride. A lot of potential riders will pass.
2. Some percentage of riders will forego the benefit of uber trips due to the higher cost. Whether it will be drunks leaving bars or whomever. Some people who would take Uber at a lower price will not take it at a higher price. Seattle may see an uptick in drunk driving as a result of this.
Governments love price control because they give the illusion of helping: uber drivers will earn more per hour. BUT - they may work fewer hours and there may be fewer Uber riders working.
In summary, this will reduce the total income of Uber drivers in Seattle. It will also reduce the number of Uber riders - some of whom are likely to become drunk drivers instead. This is bad public policy.