I don’t think that it will. Although there are more and more methods for testing drugs, the ultimate test for safety is animals because they’re most like humans.
Testing drugs in cells to see how they work is great, but a whole organism is so much more complex, and it’s hard to know how the drugs will affect all of the tissues and organs together. Right now the only way to test this is to use animals.
It would be great if one day there was a way to test drugs without animals, but I don’t know if it will happen. There are these things called “organ-on-chips” which are trying to solve this problem. They’re cool but not yet complex enough to be a replacement.